Loading [MathJax]/jax/element/mml/optable/BasicLatin.js
Research article

On the Ulam stability and existence of Lp-solutions for fractional differential and integro-differential equations with Caputo-Hadamard derivative

  • Received: 04 February 2024 Revised: 02 October 2024 Accepted: 10 November 2024 Published: 18 December 2024
  • In this paper, we investigate the existence and uniqueness of Lp-solutions for nonlinear fractional differential and integro-differential equations with boundary conditions using the Caputo-Hadamard derivative. By employing Hölder's inequality together with the Krasnoselskii fixed-point theorem and the Banach contraction principle, the study establishes sufficient conditions for solving nonlinear problems. The paper delves into preliminary results, the existence and uniqueness of Lp solutions to the boundary value problem, and presents the Ulam-Hyers stability. Furthermore, it investigates the existence, uniqueness, and stability of solutions for fractional integro-differential equations. Through standard fixed-points and rigorous mathematical frameworks, this research contributes to the theoretical foundations of nonlinear fractional differential equations. Also, the Adomian decomposition method (ADM) is used to construct the analytical approximate solutions for the problems. Finally, examples are given that illustrate the effectiveness of the theoretical results.

    Citation: Abduljawad Anwar, Shayma Adil Murad. On the Ulam stability and existence of Lp-solutions for fractional differential and integro-differential equations with Caputo-Hadamard derivative[J]. Mathematical Modelling and Control, 2024, 4(4): 439-458. doi: 10.3934/mmc.2024035

    Related Papers:

    [1] Shan Jiang, Li Liang, Meiling Sun, Fang Su . Uniform high-order convergence of multiscale finite element computation on a graded recursion for singular perturbation. Electronic Research Archive, 2020, 28(2): 935-949. doi: 10.3934/era.2020049
    [2] Li-Bin Liu, Ying Liang, Jian Zhang, Xiaobing Bao . A robust adaptive grid method for singularly perturbed Burger-Huxley equations. Electronic Research Archive, 2020, 28(4): 1439-1457. doi: 10.3934/era.2020076
    [3] Jun Pan, Yuelong Tang . Two-grid H1-Galerkin mixed finite elements combined with L1 scheme for nonlinear time fractional parabolic equations. Electronic Research Archive, 2023, 31(12): 7207-7223. doi: 10.3934/era.2023365
    [4] Hongze Zhu, Chenguang Zhou, Nana Sun . A weak Galerkin method for nonlinear stochastic parabolic partial differential equations with additive noise. Electronic Research Archive, 2022, 30(6): 2321-2334. doi: 10.3934/era.2022118
    [5] Xiu Ye, Shangyou Zhang, Peng Zhu . A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29(1): 1897-1923. doi: 10.3934/era.2020097
    [6] Leilei Wei, Xiaojing Wei, Bo Tang . Numerical analysis of variable-order fractional KdV-Burgers-Kuramoto equation. Electronic Research Archive, 2022, 30(4): 1263-1281. doi: 10.3934/era.2022066
    [7] Derrick Jones, Xu Zhang . A conforming-nonconforming mixed immersed finite element method for unsteady Stokes equations with moving interfaces. Electronic Research Archive, 2021, 29(5): 3171-3191. doi: 10.3934/era.2021032
    [8] Chunmei Wang . Simplified weak Galerkin finite element methods for biharmonic equations on non-convex polytopal meshes. Electronic Research Archive, 2025, 33(3): 1523-1540. doi: 10.3934/era.2025072
    [9] Yue Feng, Yujie Liu, Ruishu Wang, Shangyou Zhang . A conforming discontinuous Galerkin finite element method on rectangular partitions. Electronic Research Archive, 2021, 29(3): 2375-2389. doi: 10.3934/era.2020120
    [10] Yan Yang, Xiu Ye, Shangyou Zhang . A pressure-robust stabilizer-free WG finite element method for the Stokes equations on simplicial grids. Electronic Research Archive, 2024, 32(5): 3413-3432. doi: 10.3934/era.2024158
  • In this paper, we investigate the existence and uniqueness of Lp-solutions for nonlinear fractional differential and integro-differential equations with boundary conditions using the Caputo-Hadamard derivative. By employing Hölder's inequality together with the Krasnoselskii fixed-point theorem and the Banach contraction principle, the study establishes sufficient conditions for solving nonlinear problems. The paper delves into preliminary results, the existence and uniqueness of Lp solutions to the boundary value problem, and presents the Ulam-Hyers stability. Furthermore, it investigates the existence, uniqueness, and stability of solutions for fractional integro-differential equations. Through standard fixed-points and rigorous mathematical frameworks, this research contributes to the theoretical foundations of nonlinear fractional differential equations. Also, the Adomian decomposition method (ADM) is used to construct the analytical approximate solutions for the problems. Finally, examples are given that illustrate the effectiveness of the theoretical results.



    We will present a weak Galerkin finite element method for the following parabolic singularly perturbed convection-reaction-diffusion problem:

    {tuεΔu+bu+cu=f(x,y,t)inΩ×(0,T],u=0onΩ×(0,T],u(x,0)=u0in¯Ω, (1.1)

    where Ω=(0,1)2, 0<ε1, and T>0 is a constant. Assume b=b(x,y),c=c(x,y), and u0=u0(x,y) are sufficiently smooth functions on Ω, and

    b=(b1(x,y),b2(x,y))(β1,β2),c12bc0>0on Ω, (1.2)

    for some constants β1,β2, and c0. The parabolic convection-dominated problem (1.1) has been utilized in a broad range of applied mathematics and engineering including fluid dynamics, electrical engineering, and the transport problem [1,2].

    In general, the solution of the problem (1.1) will have abrupt changes along the boundary. In other words, the solution exhibits boundary/interior layers near the boundary of Ω. We are only interested in the boundary layers by excluding the interior layers which can be accomplished by assuming some extra compatible conditions on the data; see, e.g., [1,3]. The standard numerical schemes including the finite element method give unsatisfactory numerical results due to the boundary layers. Some nonphysical oscillations in the numerical solution can occur even on adapted meshes, and it is not easy to solve efficiently the resulting discrete system [4]. There are many numerical schemes for solving convection-dominated problems efficiently and accurately in the literature. These methods include Galerkin finite element methods [5,6,7], weak Galerkin finite element methods (WG-FEMs) [8,9,10], the streamline upwind Petrov-Galerkin (SUPG) [11,12], and the discontinuous Galerkin (DG) methods [13,14,15]. Among these numerical methods, the standard WG-FEM introduced in [16] is also an effective and flexible numerical algorithm for solving PDEs. Recently, the WG methods demonstrate robust and stable discretizations for singularly perturbed problems. In fact, while the WG-FEM and the hybridizable discontinuous Galerkin share something in common, the WG-FEM seems more appropriate for solving the time dependent singularly perturbed problems since the inclusion of the convective term in the context of hybridized methods is not straightforward and makes the analysis more subtle. Errors estimates of arbitrary-order methods, including the virtual element method (VEM), are typically limited by the regularity of the exact solution. A distinctive feature of the WG-FEM lies in its use of weak function spaces. Moreover, hybrid high-order (HHO) methods have similar features with WG-FEMs. In fact, the reconstruction operator in the HHO method and the weak gradient operator in WG methods are closely related, and that the main difference between HHO and WG methods lies in the choice of the discrete unknowns and the design of the stabilization operator [17]. Notably, in weak Galerkin methods, weak derivatives are used instead of strong derivatives in variational form for underlying PDEs and adding parameter free stabilization terms. Considering the application of the WG method, various PDEs arising from the mathematical modeling of practical problems in science are solved numerically via WG-FEMs using the concept of weak derivatives. There exist many papers on such PDEs including elliptic equations in [16,18,19], parabolic equations [20,21,22], hyperbolic equations [23,24], etc.

    However, to the best of the author's knowledge, there is no work regarding the uniform convergence results of the fully-discrete WG-FEM for singularly perturbed parabolic problems on layer-adapted meshes. This paper uses three layer-adapted meshes defined through mesh generating functions, namely, Shishkin-type meshes, Bakhvalov-Shishkin type meshes and Bakhvalov-type meshes given in [25]. The error estimates in this work show that one has optimal order of convergence for Bakhvalov-Shishkin type meshes and Bakhvalov-type meshes while almost optimal convergence for Shishkin-type meshes. The main ingredient of the error analysis is the use of the vertices-edges interpolation of Lin [26]. The main advantage of this interpolation operator is that we have sharper error bounds compared with the classical interpolation operators. For the sake of simplicity, the Crank–Nicolson method is used for time discretization. This scheme yields optimal order estimates for fully-discrete WG-FEM. As an alternative, one can apply a discontinuous Galerkin method in time and present optimal order estimates for the fully-discrete scheme [27].

    The rest of the paper is organized as follows. In Section 2, we introduce some notations and recall some definitions. The regularity of the solution is also summarized and three layer-adapted meshes have been introduced in Section 2. Also, we define the weak gradient and weak convection operators, and using these concepts we define our bilinear forms. In Section 3, the semi-discrete WG-FEM and its stability results have been presented. Section 4 introduces a special interpolation operator and analyses interpolation error estimates. Section 5 presents error analysis of the semi-discrete WG-FEM for the problem (1.1) on the layer-adapted meshes. In Section 6, we apply the Crank-Nicolson scheme on uniform time mesh in time to obtain the fully-discrete WG-FEM, and prove uniform error estimates on the layer-adapted meshes. In Section 7, we conduct some numerical examples to validate the robustness of the WG-FEM for the problem (1.1). Summary on the contributions of this work are presented in Section 8.

    Let S be a measurable subset of Ω. We shall use the classical Sobolev spaces Wr,q(S),Hr(S)=Wr,2(S),Hr0(S),Lq(S)=W0,q(S) for negative integers r>0 and 1q, and (,)S for the L2 inner product on S. The semi-norm and norm on Hr(S) are denoted by ||r,S and r,S, respectively. If S=Ω, we do not write S in the subscript. Throughout the study, we shall use C as a positive generic constant, which is independent of the mesh parameters h and ε.

    This section deals with the introduction of a decomposition of the solution which provides a priori information on the exact solution and its derivatives. Based on this solution decomposition, we construct layer-adapted meshes. As we noted in the introduction, the solution of (1.1) exhibits typically two exponential boundary layers at x=1 and y=1. The following lemma gives some information on the solution decomposition and bounds on the solution of (1.1) and its derivatives.

    Lemma 2.1. Let k be positive integer and l(0,1). Assume that the solution u of the problem (1.1) belongs to the space Ck+l(QT) where QT:=Ω×(0,T]. Assume further that the solution u can be decomposed into a smooth part uR and layer components uL0,uL1, and uL1 with

    u=uR+uL,uL=uL0+uL1+uL2,(x,y)¯Ω, (2.1)

    where the smooth and layer parts satisfy

    |i+j+ruRixjytr(x,y)|C (2.2)
    |i+j+ruL0ixjytr(x,y)|Cεieβ1(1x)/ε, (2.3)
    |i+j+ruL1ixjytr(x,y)|Cεjeβ2(1y)/ε, (2.4)
    |i+j+ruL2ixjytr(x,y)|Cε(i+j)eβ1(1x)/εeβ2(1y)/ε, (2.5)

    for any (x,y)¯Ω,t[0,T], and positive integers i,j,r with i+j+2rk, and C only depends on b,c, and f and is independent of ε. Here, uR is the regular part of u, uL0 is an exponential boundary layer along the side x=1 of Ω, uL1 is an exponential boundary layer along the side y=1, while uL2 is an exponential corner layer at (1,1).

    Proof. Under some smoothness conditions and strong imposed compatibility requirements, Shishkin proved this solution decomposition; see, [1].

    Let Nx and Ny be positive integers. For the sake of simplicity, we assume that Nx=Ny=N is an even integer number. We shall construct the tensor product mesh TN={Tij}i,j=1,2,N in ¯Ω consisting of elements Tij=Ii×Kj with the intervals Ii=(xi1,xi) and Kj=(yj1,yj), where the mesh points are defined by

    0=x0<x1<,xNx=1,0=y0<y1<,yNy=1.

    Since the construction of the meshes in both directions is similar, the mesh construction in x-variable is given here.

    We define the transition parameter as

    τ=min

    where \sigma\geqslant p+1 is a positive constant. Here, p is the degree of the polynomials used in the approximation space. The function \varphi obeys the conditions

    \begin{align} \varphi(0) = 0, \; \varphi^\prime > 0, \; \; \varphi^{\prime \prime}\geq 0. \end{align} (2.6)

    Assumption 1. Throughout this article, we assume that \varepsilon \leq C N^{-1} such that \tau = \frac{(p+1) \varepsilon}{\beta_{1}} \varphi(1/2) , since otherwise the analysis can be carried out using uniform mesh.

    Let the mesh points x_i be equally distributed in [0, x_{N/2}] with N/2 intervals and distributed [x_{N/2}, 1] with N/2 intervals using the mesh generating function defined by

    \begin{align} x_i = \lambda(i/N) = \begin{cases} 2(1-\tau)i/N, &\text{for } i = 0, 1, \dots, N/2, \\ 1-\frac{(k+1)\varepsilon}{\beta_1}\varphi(1-i/N), &\text{for } i = N/2, N/2 +1, \dots, N. \end{cases} \end{align} (2.7)

    For example, as in [25], the Shishkin-type (S-type) meshes can be deduced by \varphi(1/2) = \ln N while Bakhvalov-type meshes (B-type) can be recovered by taking \varphi(1/2) = \ln (1/\varepsilon) .

    We will use the mesh characterizing function \psi defined by \psi = e^{-\varphi} , which is an essential tool in our analysis below.

    Following [2], we list some famous adaptive meshes including S-type, Bakhvalov-Shishkin meshes (BS-mesh), and B-type in Table 1.

    Table 1.  Frequently used layer-adapted meshes.
    S-type BS B-type
    \varphi(t) 2 t \ln N -\ln \left[1-2\left(1-N^{-1}\right) t\right] -\ln [1-2(1-\varepsilon) t]
    \psi(t) N^{-2 t} 1-2\left(1-N^{-1}\right) t 1-2(1-\varepsilon) t
    \max \left|\psi^{\prime}\right| C \ln N C C

     | Show Table
    DownLoad: CSV

    Similarly, we define the transition point in the y -direction as

    \tau_y: = \min \left(\frac{1}{2}, \frac{\sigma \varepsilon}{\beta_{2}} \varphi(1/2)\right).

    We first split the domain \varOmega into four subdomains as in Figure 1:

    \begin{align*} \varOmega_r&: = [0, 1- \tau]\times [0, 1-\tau_y], \qquad \varOmega_x: = [1- \tau, 1]\times [0, 1-\tau_y], \\ \varOmega_y&: = [0, 1- \tau]\times [1-\tau_y, 1], \qquad \varOmega_{xy}: = [1- \tau, 1]\times [1-\tau_y, 1]. \end{align*}
    Figure 1.  Tensor product Shishkin mesh for N = 8 .

    Clearly, the mesh is uniform in \varOmega_r with a mesh size of \mathcal{O}(N^{-1}) , highly anisotropic in \varOmega_x and \varOmega_y , while it is very fine in \varOmega_{xy} .

    Let h_i^x: = x_i-x_{i-1}, \; {i = 1, \dots, N}\quad h_{{j}}^y: = y_j-y_{j-1}, \; {j = 1, \dots, N} be the mesh sizes of the subintervals. For the sake of simplicity, we assume that \beta_1 = \beta_2 = \beta . Then, one has h_i^x = h_{{j}}^y , and we simply write h_i, \; i = 1, \dots, N for simplicity. The following technical lemmas show the smallness of the boundary layer-functions and the basic properties of the mesh sizes of the layer-adapted meshes.

    Lemma 2.2. [28] Denote by \Theta_i = \min \left\{h_i / \varepsilon, 1\right\} e^{-\alpha\left(1-x_i\right) / \sigma \varepsilon} for i = N / 2+1, \ldots, N . There exists a constant C > 0 independent of \varepsilon and N such that

    \max\limits_{N / 2+1 \leq i \leq N} \Theta_i \leq C N^{-1} \max \left|\psi^{\prime}\right|
    \sum\limits_{i = N / 2+1}^N \Theta_i \leq C

    Lemma 2.3. [28] For the layer-adapted meshes we considered here, we have

    h_1 = h_2 = \ldots = h_{N / 2} \mathit{\text{and}} { \min\limits_{i = 1, \dots, N}h_i }\geq C \varepsilon N^{-1} \max \left|\psi^{\prime}\right|.

    Moreover, for i = N / 2+1, \cdots, N,

    \begin{aligned} \begin{array}{ll} h_i = 2\tau/N, & \mathit{\text{for S-type}} \\ 1 \geq \frac{h_{i+1}}{h_i} \geq C, \quad & \mathit{\text{for BS- mesh}} \end{array} \end{aligned}

    and, for the B-type mesh,

    \begin{aligned} \begin{array}{ll} i = N / 2+2, \cdots, N, \quad 1 \geq \frac{h_{i+1}}{h_i} \geq C, \\ i = 1, 2, \cdots, N / 2, \quad h_{N / 2+i} \geq \frac{\sigma \varepsilon}{\beta} \frac{1}{i+1}, \\ \end{array} \end{aligned}

    where C > 0 is a constant independent of \varepsilon and N .

    A weak formulation of the problems (1.1) and (1.2) is to look for u\in H_0^1(\varOmega) such that

    \begin{align} (u_t, v) +A(u, v) = (f, v)\quad \forall v\in H_0^1(\varOmega), \end{align} (2.8)

    where the bilinear from {A}(\cdot, \cdot) is defined by

    \begin{align*} A(u, v): = \varepsilon(\nabla u, \nabla v)+ ({\bf b}\cdot \nabla u, v)+(cu, v). \end{align*}

    Based on the weak formulation (2.8), we define the WG-FEM on the layer-adapted mesh. Let p be a positive integer. We define a local WG-FE space V(p, K) on each K\in \mathcal{T}_N given by

    \begin{align*} V(p, K): = \left\{ v_N = \left\{v_{0}, v_{b}\right\}:\left.v_{0}\right|_{K} \in \mathbb{Q}_{p}(K), \left.v_{b}\right|_{e} \in \mathbb{P}_{p}(e), e \subset \partial K\right\}, \end{align*}

    where \mathbb{Q}_p(K) is the polynomials of degree p on K in both variables, and \mathbb{P}_p(e) denotes the polynomials of degree p on the edge e .

    Defining the WG finite element space V_N globally on \mathcal{T}_N as

    \begin{align} V_{N} = \left\{v_N = \left\{v_{0}, v_{b}\right\}: \left.v_{N}\right|_{K}\in V(p, K), \left.v_{b}\right|_{e\cap \partial K_1} = \left.v_{b}\right|_{e\cap \partial K_2}, \partial K_1 \cap \partial K_2 = \{e\}\right\}, \end{align} (2.9)

    and its subspace

    V_{N}^{0} = \left\{v: v \in V_{N}, v_{b} = 0 \text { on } \partial \Omega\right\}.

    The weak gradient operator \nabla_w u_N \in [\mathbb{Q}_{p-1}(K)]^2 can be defined on K as

    \begin{align} (\nabla_w u_N, \psi)_K = -(u_0, \nabla\cdot \psi)_K+\langle u_b, \psi\cdot {\bf n}\rangle_{\partial K}\quad \quad \forall \psi \in [\mathbb{Q}_{p-1}(K)]^2, \end{align} (2.10)

    where {\bf n} represents the outward unit normal \partial K , (w, v)_K denotes the inner product on K for functions w and v , and \langle w, v\rangle_{\partial K} is the L^{2}- inner product on \partial K .

    The weak convection operator {\bf b}\cdot \nabla_w u_N \in \mathbb{Q}_{p}(K) can be defined on K as

    \begin{align} ( {\bf b}\cdot \nabla_w u_N, \xi)_K = -(u_0, \nabla\cdot ( {\bf b} \xi))_K+\langle u_b, {\bf b}\cdot {\bf n} \xi \rangle_{\partial K}\quad \quad \forall \xi \in \mathbb{Q}_{p}(K). \end{align} (2.11)

    For simplicity, we adapt

    \begin{align*} \big(\phi, \psi\big)& = \sum\limits_{K\in \mathcal{T}_N}\big(\phi, \psi\big)_K, \quad \Vert \phi\Vert^2 = \big(\phi, \phi\big), \quad \big\langle \phi, \psi\big\rangle = \sum\limits_{K \in \mathcal{T}_N} \big\langle \phi, \psi \big\rangle_{\partial K} . \end{align*}

    For u_N = \{u_0, u_b\}\in V_N and v_N = \{v_0, v_b\}\in V_N , the bilinear form {\cal A}_{w}(\cdot, \cdot) is given by

    \begin{align} &&{\cal A}_{w}(u_N, v_N) = \varepsilon \big( \nabla_w u_N, \nabla_w v_N\big)+\big( {\bf b}\cdot \nabla_w u_N, v_0\big)+(cu_0, v_o)\\ &&+{\cal S}_d(u_N, v_N)+{\cal S}_c(u_N, v_N), \end{align} (2.12)

    where s_d(\cdot, \cdot) and s_c(\cdot, \cdot) are bilinear forms defined by

    \begin{align*} {\cal S}_d(u_N, v_N)& = \sum\limits_{K\in \mathcal{T}_N}{\rho_K} \big\langle u_0-u_b, v_0-v_b\big\rangle_{\partial K}, \\ {\cal S}_c(u_N, v_N)& = \sum\limits_{K\in \mathcal{T}_N}\big\langle {\bf b} \cdot {\bf n} (u_0-u_b), v_0-v_b\big\rangle_{\partial K^{+}} \end{align*}

    with \partial K^{+} = \{ z \in \partial K: {\bf b}(z) \cdot {\bf n}(z) \geq 0\} and {\rho_K} is the penalty term given by

    \begin{align} \rho_K: = \begin{cases} 1, & \text{if } K\subset \varOmega_r, \\ N{{(\max \left|\psi^{\prime}\right|)^{-1}}}, & \text{if } K\subset \varOmega\setminus \varOmega_r. \end{cases} \end{align} (2.13)

    Given a mesh rectangle K , its dimensions parallel to the x and y -axes are written as h_{x, K} and h_{y, K} , respectively.

    Lemma 2.4. [29] For all K \in {\cal T}_{N} with h_K = \min \{h_{x, K}, h_{y, K}\} , there exists a constant C depending only on p such that

    \begin{align} \Vert u_N\Vert _{L^2(\partial K)}^2\le C h_K^{-1}\Vert u_N\Vert _{L^2(K)}^2, \quad \forall u_N\in \mathbb {P}_p(K). \end{align} (2.14)

    We next formulate our semi-discrete WG scheme as follows (Algorithm 1).

    Algorithm 1 The semi-discrete WG-FEM for problem (1.1)
    Find u_N = (u_0, u_b)\in V_N^0 such that
              \begin{align} & (u_{0}^\prime, v_0) + {\cal A}_{w}(u_N, v_N) = (f, v_0)\quad \forall v_N = (v_0, v_b)\in V_N^0, \\ &u_N(0) = u^0(0), \end{align}\ \ \ \ \ (2.15)
    where u^0(0)\in V_N^0 is an approximation of u(0) .

    This section is devoted to establishing the stability results of the WG-FEM defined by (2.15). Define the energy norm \Vert \cdot\Vert_E on the weak function space V_N for v_N = \{v_0, v_b\}\in V_N ,

    \begin{align} &&\Vert v_N \Vert_E^{2}: = \varepsilon \sum\limits_{K \in \mathcal{T}_{N}}\Vert\nabla_{w} v_N\Vert_{K}^{2}+\sum\limits_{K \in \mathcal{T}_{N}}\Vert|\mathbf{b} \cdot \mathbf{n}|^{1 / 2}\left(v_{0}-v_{b}\right)\Vert_{\partial K}^{2}\\&&+\left\Vert v_{0}\right\Vert^{2}+{\cal S}_{d}(v_N, v_N). \end{align} (3.1)

    Define also an H^1 equivalent norm on V_N by

    \begin{align} \Vert v_N\Vert_{\varepsilon}^2: = \varepsilon \sum\limits_{K \in \mathcal{T}_{N}}\Vert\nabla v_0\Vert_{K}^{2}+\sum\limits_{K \in \mathcal{T}_{N}}\Vert|\mathbf{b} \cdot \mathbf{n}|^{1 / 2}\left(v_{0}-v_{b}\right)\Vert_{\partial K}^{2}+\left\Vert v_{0}\right\Vert^{2}+{\cal S}_{d}(v_N, v_N). \end{align} (3.2)

    The equivalence of these two norms on V_N^0 is given in the next lemma.

    Lemma 3.1. For v_N\in V_N^0 , one has

    \begin{align*} C \Vert v_N\Vert_\varepsilon \leq \Vert v_N\Vert _E \leq C \Vert v_N\Vert_\varepsilon. \end{align*}

    Proof. For v_N = \{v_0, v_b\}\in V_N^0 , it follows from the weak gradient operator (2.10) and integration by parts that;

    \begin{align} \big( \nabla_w v_N, w\big)_K = \big(\nabla v_0, w\big)_K-\big\langle v_0-v_b, {\bf n} \cdot w\big\rangle_{\partial K}, \;\;\forall\; w\in [\mathbb{Q}_{p-1}(K)]^2, \forall K\in \mathcal{T}_N. \end{align} (3.3)

    Choosing w = \nabla_w v_N in (3.3) and using the Cauchy-Schwartz inequality and the trace inequality (4), we arrive at

    \begin{align*} \big( \nabla_w v_N, \nabla_w v_N\big)_K & = \big(\nabla v_0, \nabla_w v_N \big)_K-\big\langle v_0-v_b, {\bf n} \cdot \nabla_w v_N \big\rangle_{\partial K}\\ &\le \Vert \nabla v_0\Vert_{L^2(K)} \Vert \nabla_w v_N\Vert_{L^2(K)}+ \Vert v_0-v_b\Vert_{L^2(\partial K} \Vert \nabla_w v_N\Vert_{L^2(\partial K)}\\ &\le \big( \Vert \nabla v_0\Vert_{L^2(K)}+Ch_K^{-1/2} \Vert v_0-v_b\Vert_{L^2(\partial K)}\big) \Vert \nabla_w v_N\Vert_{L^2( K)}. \end{align*}

    Hence, we get

    \begin{align*} \Vert \nabla_w v_N\Vert_{L^2(K)}\leq \Vert \nabla v_0\Vert_{L^2(K)}+Ch_K^{-1/2} \Vert v_0-v_b\Vert_{L^2(\partial K)}. \end{align*}

    Summing over all K\in \mathcal{T}_N yields

    \begin{align*} \varepsilon \Vert \nabla_w v_N\Vert^2\leq 2\Big(\varepsilon \Vert \nabla v_0\Vert^2+C\sum\limits_{K\in \mathcal{T}_N} \varepsilon h_K^{-1} \Vert v_0-v_b\Vert^2_{L^2(\partial K)}\Big). \end{align*}

    From the penalty term (2.13), we get

    \begin{align*} \cfrac{\varepsilon h_K^{-1}}{\rho_K}\le C, \quad \forall K\in \mathcal{T}_N, \end{align*}

    and

    \begin{align*} \sum\limits_{K\in \mathcal{T}_N}\varepsilon h_K^{-1} \Vert v_0-v_b\Vert_{L^2(\partial K)} = \sum\limits_{K\in \mathcal{T}_N} \cfrac{\varepsilon h_K^{-1}}{\rho_K} \rho_K \Vert v_0-v_b\Vert_{L^2(\partial K)}\leq C s_d(v_N, v_N). \end{align*}

    As a result, for v_N\in V_N^0 , we have

    \begin{equation} \Vert v_N\Vert_E\leq C \Vert v_N\Vert_\varepsilon. \end{equation} (3.4)

    Taking w = \nabla v_0 in (3.3) and using the Cauchy-Schwartz inequality, we get

    \begin{align*} \big(\nabla v_0 , \nabla v_0\big)_K& = (\nabla_w v_N, \nabla v_0)_K+ \big\langle v_0-v_b, {\bf n} \cdot \nabla v_0\big\rangle_{\partial K}\\ &\le \Vert \nabla_w v_N\Vert_{L^2(K)} \Vert \nabla v_0\Vert_{L^2(K)}+ \Vert v_0-v_b\Vert_{L^2(\partial K} \Vert \nabla v_0\Vert_{L^2(\partial K)}\\ &\le \big( \Vert \nabla_w v_N\Vert_{L^2(K)}+Ch_K^{-1/2} \Vert v_0-v_b\Vert_{L^2(\partial K)}\big) \Vert \nabla v_0\Vert_{L^2( K)}, \end{align*}

    where we have again used the trace inequality (4).

    Consequently,

    \begin{align*} \Vert \nabla v_0\Vert_{L^2(K)}\leq \Vert \nabla_w v_N\Vert_{L^2(K)}+Ch_K^{-1/2} \Vert v_0-v_b\Vert_{L^2(\partial K)}. \end{align*}

    Summing over all K\in \mathcal{T}_N yields

    \begin{align*} \varepsilon \Vert \nabla v_0\Vert^2\leq 2\Big(\varepsilon \Vert \nabla_w v_N\Vert^2+C\sum\limits_{K\in \mathcal{T}_N} \varepsilon h_K^{-1} \Vert v_0-v_b\Vert^2_{L^2(\partial K)}\Big). \end{align*}

    Therefore, we have

    \begin{align*} \varepsilon \Vert \nabla v_0\Vert^2\leq 2 \big( \varepsilon \Vert \nabla_w v_N\Vert ^2+C s_d(v_N, v_N)\big), \end{align*}

    which implies

    \begin{equation} \Vert v_N\Vert_\varepsilon\le C \Vert v_N\Vert_E. \end{equation} (3.5)

    From (3.4) and (3.5), we obtain the desired conclusion, which completes the proof.

    We shall show the coercivity of the bilinear form {\cal A}_{w}(\cdot, \cdot) in \Vert \cdot \Vert_E norm on V_N^0 .

    Lemma 3.2. For any v_N\in V_N^0 , the following inequality holds:

    \begin{align} {\cal A}_{w}(v_N, v_N)\ge C \Vert v_N\Vert_E^2, \quad \forall v_N\in V_N^0. \end{align} (3.6)

    Proof. For v_N = \{v_0, v_b\}, w_N = \{w_0, w_b\}\in V_N^0 , using the weak convection derivative (2.11) and integration by parts gives

    \begin{align} \begin{split} \left(\boldsymbol{b} \cdot \nabla_{w} v_N, w_0\right)& = -(v_0, \nabla \cdot(\boldsymbol{b} w_0))+\big\langle v_b, \boldsymbol{b} \cdot \boldsymbol{n} w_0\big \rangle \\ & = (\boldsymbol{b} \cdot \nabla v_0, w_0)-\big\langle\boldsymbol{b} \cdot \boldsymbol{n}(v_0-v_b), w_0\big \rangle \end{split} \end{align} (3.7)

    and

    \begin{align} \begin{split} \left(\boldsymbol{b} \cdot \nabla_{w} w_N, v_0\right){}& = -(w_0, \nabla \cdot(\boldsymbol{b} v_0))+\big\langle w_b, \boldsymbol{b} \cdot \boldsymbol{n} v_0\big\rangle \\ & = -(w_0, \nabla \cdot(\boldsymbol{b} v_0))+\big\langle w_b, \boldsymbol{b} \cdot \boldsymbol{n}(v_0-v_b)\big\rangle \end{split} \end{align} (3.8)

    where we use the facts that v_N, w_N\in V_N^0 , and \big \langle\boldsymbol{b} \cdot \boldsymbol{n}v_b, w_b\big \rangle = 0 in the last inequality.

    Combining (3.7) and (3.8), and taking v_N = w_N , we obtain

    \begin{align} \left(\boldsymbol{b} \cdot \nabla_{w } v_N, v_0\right) = -\frac{1}{2}(\nabla \cdot \boldsymbol{b} v_0, v_0) -\frac{1}{2}\big\langle\boldsymbol{b} \cdot \boldsymbol{n}(v_0-v_b), v_0-v_b\big \rangle . \end{align} (3.9)

    From (3.9), we have

    \begin{align*} {\cal A}_{w}(v_N, v_N) = &\varepsilon\left(\nabla_{w} v_N, \nabla_{w} v_N\right)+\left(\left(-\frac{1}{2} \nabla \cdot \boldsymbol{b}\right) v_0, v_0\right)+{\cal S}_d(v_N, v_N)+{\cal S}_c(v_N, v_N) \\ &-\frac{1}{2}\big\langle\boldsymbol{b} \cdot \boldsymbol{n}(v_0-v_b), v_0-v_b\rangle \\ \geq & \varepsilon \Vert \nabla_w v_N\Vert^2+ c_0 \Vert v_0\Vert^2+\frac{1}{2} \sum\limits_{T \in \mathcal{T}_{N}}\Big\Vert \vert {\bf b} \cdot {\bf n}\vert^2(v_0-v_b) \Big\Vert^2_{\partial K} +{\cal S}_d(v_N, v_N)\\ \geq & C \Vert v_N\Vert_E^2, \end{align*}

    which yields (3.6) with C = \min \left\{c_{0}, \frac{1}{2}\right\} . The proof is completed.

    Therefore, the existence of a unique solution to (2.15) follows from the coercivity property of the bilinear form (3.6). As a result of the two lemmas above, the bilinear form is also coercive in the \Vert \cdot \Vert _\varepsilon -norm in the sense that for any v_N\in V_N^0 , there holds

    \begin{align} {\cal A}_{w}(v_N, v_N) \ge C \Vert v_N\Vert^2_\varepsilon. \end{align} (3.10)

    Lemma 3.3. If f \in {L}^{2}(\Omega) for each t\in (0, T) , then there is a constant C > 0 independent of t and mesh size h such that the solution u_{N}(t) = \{u_0(t), u_b(t)\} defined in (2.15) satisfies

    \begin{equation} \|u_0(t)\|^{2}\leq C\|u^0\|^{2}+\int_{0}^{t}\|f(s)\|^{2}ds, \; \forall\; t\in(0, T]. \end{equation} (3.11)

    Proof. Choosing v = u_{N}(t) in (2.15) gives that

    \begin{equation*} \frac{1}{2}\frac{d}{dt}\|u_0(t)\|^{2}+{\cal A}_{w}(u_{N}, u_{N}) = (f, u_0). \end{equation*}

    Using the Cauchy-Schwarz inequality and the coercivity (3.6) of the bilinear form {\cal A}_{w}(\cdot, \cdot) ,

    \begin{align*} \frac{d}{dt}\|u_0(t)\|^{2} &\leq 2 (f, u_0)\\ &\leq \|f\|^{2}+\|u_0\|^{2}. \end{align*}

    Integrating the above inequality with respect to the time variable t , we arrive at

    \begin{equation} \|u_0(t)\|^{2}\leq \|u^0\|^{2}+\int_{0}^{t} \|f(s)\|^{2} ds+\int_{0}^{t}\|u_0(s)\|^{2}ds. \end{equation} (3.12)

    Using the Gronwall's inequality gives the desired conclusion. We are done.

    First, we define "vertices-edges' interpolation \mathcal{P}_K v of a function v on an element K as follows. Let \hat{K}: = (-1, 1)\times (-1, 1) be the reference element with the vertices \hat{a_i} and the edges \hat{e}_i for i = 1, \dots, 4 . For \hat{v}(\cdot, \cdot) \in C(\hat{K}) , the approximation \hat{\mathcal{P}}: C(\hat{K})\to \mathbb{Q}_p(\hat{K}) is given by

    \begin{align} &( \hat{\mathcal{P}} \hat{v})\left(\hat{a}_{i}\right) = \hat{v}\left(\hat{a}_{i}\right) \quad \text { for } i = 1, \ldots, 4, \end{align} (4.1)
    \begin{align} &\int_{\hat{e}_{i}}( \hat{\mathcal{P}} \hat{v}) q d s = \int_{\hat{e}_{i}} \hat{v} q d s \quad \forall q \in \mathbb{P}_{p-2}\left(\hat{e}_{i}\right) \text { for } i = 1, \ldots, 4, \end{align} (4.2)
    \begin{align} &\iint_{\hat{K}}( \hat{\mathcal{P}} \hat{v}) q d \xi d \eta = \iint_{\hat{K}} \hat{v} q d \xi d \eta \quad \forall\; q \in \mathbb{Q}_{p-2}(\hat{K}). \end{align} (4.3)

    The approximation operator \hat{\mathcal{P}} is well-defined [30]. Thus, we can define a continuous global interpolation operator \mathcal{P}_N: C(\overline{\varOmega})\to V_N by writing

    \begin{align} ( \mathcal{P}_N u)|_{K}: = ( \hat{\mathcal{P}}(u\circ G_K))\circ G_K^{-1}\qquad \forall\; K \in \mathcal{T}_N, u\in C(\overline{\varOmega}), \end{align} (4.4)

    where the bijective mapping G_K:\hat{K}\to K is given by G_K(\xi, \eta) = (x, y) = (x_K+h_K^x \xi/2, y_K+h_K^y \eta/2) .

    This interpolation operator has the following stability estimate [30]

    \begin{align} \Vert \mathcal{P}_N \phi\Vert_{L^\infty(K)}\leq C\Vert \phi\Vert_{L^\infty(K)}, \quad \forall \phi\in C(K). \end{align} (4.5)

    Since our approximation operator (\mathcal{P}_N v)|_{K} is continuous on K , we have \{ (\mathcal{P}_N v)|_{K}, (\mathcal{P}_N v)|_{e} \} \in V_N for e\in \partial K . By the trace theorem, we will denote this by \mathcal{P}_N v .

    Lemma 4.1. [31] For any \phi\in H^1(K) ,

    \begin{align*} \big(\nabla_w ( \mathcal{P}_N \phi), \psi\big) = \big(\nabla ( \mathcal{P}_N \phi), \psi\big) \quad \forall \psi \in \mathbb{Q}_{p-1}(K), K\in \mathcal{T}_N. \end{align*}

    We recall some technical results from [30].

    Lemma 4.2. For any K \in \mathcal{T}_N and q \in[1, \infty] , there exists a constant C such that the vertices-edges-element approximant \mathcal{P}_{N}\phi satisfies

    \left\|v- \mathcal{P}_{N}\phi\right\|_{L^q(K)} \leq C \sum\limits_{i+j = p+1} h_{x, K}^i h_{y, K}^j\left\|\frac{\partial^{p+1}\phi}{\partial x^i \partial y^j}\right\|_{L^q(K)}

    for all \phi \in W^{p+1, q}(K) .

    Lemma 4.3. Let s \in [1, p] . The following estimates hold for any K \in \mathcal{T}_{N} :

    \begin{aligned} &\left\|\left(\psi-{ \mathcal{P}_{N} }\psi\right)_{x}\right\|_{L^2(K)} \leq C \sum\limits_{i = 0}^s h_{x, K}^{i} h_{y, K}^{s-i}\left\|\frac{\partial^{s+1} \psi}{\partial x^{i+1} \partial y^{s-i}}\right\|_{L^2(K)} , \\ &\left\|\left(\psi-{ \mathcal{P}_{N} } \psi\right)_{y}\right\|_{L^2(K)} \leq C \sum\limits_{i = 0}^s h_{x, K}^{i} h_{y, K}^{s-i}\left\|\frac{\partial^{s+1} \psi}{\partial x^{i} \partial y^{s-i+1}}\right\|_{L^2(K)} \end{aligned}

    for all \psi \in H^{r+1}(K) .

    A careful inspection of the proof of Lemma 4.3 in [30] reveals that the following results also hold true.

    Lemma 4.4. For K \in \mathcal{T}_{N} and s \in [1, p+1-\ell] with \ell = 1, 2, \dots, p+1 , there exists a constant C such that the vertices-edges-element approximant \mathcal{P}_{N}\phi satisfies

    \begin{aligned} &\left\|\cfrac{\partial^\ell(\phi- \mathcal{P}_{N}\phi)}{\partial x^\ell}\right\|_{L^2(K)} \leq C \sum\limits_{i = 0}^s h_{x, K}^{i} h_{y, K}^{s-i}\left\|\frac{\partial^{r+\ell}\phi}{\partial x^{i+\ell} \partial y^{s-i}}\right\|_{L^2(K)}, \\ &\left\|\cfrac{\partial^\ell (\phi- \mathcal{P}_{N}\phi)}{\partial y^\ell}\right\|_{L^2(K)} \leq C \sum\limits_{i = 0}^s h_{x, K}^{i} h_{y, K}^{s-i}\left\|\frac{\partial^{r+\ell}\phi}{\partial x^{i} \partial y^{s-i+\ell}}\right\|_{L^2(K)} \end{aligned}

    for all \phi \in H^{r+1}(K) .

    Lemma 4.5. Let the assumption of Lemma 2.1 hold such that u = u_R+u_L, \; u_L = u_{L_0}+u_{L_1}+u_{L_2}. Then there is a constant C > 0 such that the following interpolation error estimates are satisfied:

    \begin{align} &\|u_R- \mathcal{P}_{N} u_R\|_{L^2(\Omega)} \leq C N^{-(p+1)}, \end{align} (4.6)
    \begin{align} & \|u_L\|_{L^2(\Omega_{r})}\leq C \varepsilon^{1 / 2} N^{-(p+1)}, \end{align} (4.7)
    \begin{align} & \|\nabla u_{L}\|_{L^{2}(\Omega_{r})} \leq C \varepsilon^{-1/2}N^{-(p+1)}, \end{align} (4.8)
    \begin{align} & \|\Delta u_{L}\|_{L^{2}(\Omega_{r})} \leq C \varepsilon^{-3/2}N^{-(p+1)}, \end{align} (4.9)
    \begin{align} & N^{-1} \|\nabla \mathcal{P}_{N} u_{L}\|_{L^2(\Omega_{r})}+ \| \mathcal{P}_{N} u_{L}\|_{L^2(\Omega_{r})}\leq C N^{-(p+3/2)}, \end{align} (4.10)
    \begin{align} &\|u_{L}- \mathcal{P}_{N} u_{L}\|_{L^2(\Omega)} \leq C(N^{-1} \max \left|\psi^{\prime}\right|)^{p+1}. \end{align} (4.11)

    Proof. The proof of (4.6) follows from Lemma 4.2 and the solution decomposition (2.2) of Lemma 2.1.

    Using the decay bound (2.3) of u_{L_0} and the fact that \varphi(1/2)\geq \ln N , we have

    \begin{align*} \Vert u_{L_0}\Vert^2_{L^2(\varOmega_r)}& = \int_0^{1-\tau}\int_0^{1-\tau}\vert u_{L_0}\vert^2 d x dy\\&\leq C \int_0^{1-\tau}\int_0^{1-\tau}e^{-2\beta_1(1-x)/\varepsilon} d x dy\leq C \varepsilon N^{-2(p+1)}, \end{align*}

    which shows (4.7) for u_{L_0} . Similar arguments can be applied to the layer functions u_{L_1} and u_{L_2} . Thus, we complete the proof of (4.7). For (4.8) and (4.9), we will prove for u_{L_0} since the other two parts follow similarly. One can use the decay bound (2.3) of u_{L_0} to obtain

    \begin{align*} \begin{aligned} \left\|\nabla u_{L_0}\right\|_{L^2\left(\Omega_{r}\right)}^2 & \leq C \varepsilon^{-2} \int_0^{1-\tau} \int_0^{1- \tau} e^{-2\beta_1(1-x) / \varepsilon} d x d y \\ & \leq C \varepsilon^{-1} e^{-2\beta_1 \tau / \varepsilon} \\ & \leq C \varepsilon^{-1} N^{-2(p+1)}. \end{aligned} \end{align*}

    We now shall prove (4.9). Appealing (2.3), one gets

    \begin{align*} \begin{aligned} \left\|\Delta u_{L_0}\right\|_{L^2\left(\Omega_{r}\right)}^2 & \leq C \varepsilon^{-4} \int_0^{1-\tau} \int_0^{1- \tau} e^{-2\beta_1(1-x) / \varepsilon} d x d y \\ & \leq C \varepsilon^{-3} e^{-2\beta_1 \tau / \varepsilon} \\ & \leq C \varepsilon^{-3} N^{-2(p+1)}. \end{aligned} \end{align*}

    The proof of (4.10) is a little longer. Using an inverse estimate yields

    N^{-1} \|\nabla \mathcal{P}_{N} u_{L}\|_{L^2(\Omega_{r})}+ \| \mathcal{P}_{N} u_{L}\|_{L^2(\Omega_{r})}\leq C\| \mathcal{P}_{N} u_{L}\|_{L^2(\Omega_{r})}.

    Hence, we shall estimate \| \mathcal{P}_{N} u_{L}\|_{L^2(\Omega_{r})} . With the help of the stability estimate (4.5) and the decay bound (2.3) of u_{L_0} , we have

    \begin{align*} \begin{aligned} \left\| \mathcal{P}_N u_{L_0}\right\|_{L^2\left(\Omega_{r}\right)}^2 & \leq C \int_0^{1-\tau} \sum\limits_{i = 1}^{N/2}\int_{x_{i-1}} ^{x_i}e^{-2\beta_1(1-x_i) / \varepsilon} d x d y . \end{aligned} \end{align*}

    If i < N / 2 , then the sum can be small as a function of \varepsilon but not small if i = N / 2 . For i = 1, \ldots, N / 2-1 and x \in\left[x_{i-1}, x_i\right] , we have

    e^{-2 \beta_1\left(1-x_i\right) / \varepsilon} = e^{2 \beta_1\left(x_{N / 2}-x_{N / 2-1}\right) / \varepsilon} e^{-2 \beta_1\left(1-x_{i-1}\right) / \varepsilon} \leq e^{2 \beta_1\left(x_{N / 2}-x_{N / 2-1}\right) / \varepsilon} e^{-2 \beta_1(1-x) / \varepsilon}

    and when i = N / 2 , again using the fact \varphi(1/2)\geq \ln N ,

    e^{-2 \beta_1\left(1-x_{N / 2}\right) / \varepsilon} = e^{-2 \beta_1 \tau / \varepsilon}\leq N^{-2(p+1)}.

    Thus,

    \begin{aligned} \left\| \mathcal{P}_N u_{L_0}\right\|_{L^2( \Omega_{r})}^2 & \leq C e^{2 \beta_1\left(x_{N / 2}-x_{N / 2-1}\right) / \varepsilon} \int_0^{x_{N / 2-1}} e^{-2 \beta_1(1-x) / \varepsilon} d x+C N^{-2p-3} \\ & \leq C \varepsilon e^{-2 \beta_1\left(1-x_{N / 2}\right) / \varepsilon}+C N^{-2p-3} \\ & \leq C\left(\varepsilon N^{-2(p+1)}+ N^{-2p-3}\right) \end{aligned}

    which proves (4.10). To prove (4.11), we use (4.7) and (4.10) to obtain

    \left\|u_L- \mathcal{P}_N u_L\right\|_{L^2( \Omega_{r})} \leq C\left(\varepsilon^{1 / 2} N^{-(p+1)}+N^{-p-3/2}\right) \leq C (N^{-1} \max \left|\psi^{\prime}\right|)^{p+1}.

    On the set \varOmega_r\cup \varOmega_y , from the triangle inequality, one obtains

    \begin{align*} \left\|u_{L_0}- \mathcal{P}_N u_{L_0}\right\|_{L^2(\varOmega_r\cup \varOmega_y)}&\leq C(\left\|u_{L_0}\right\|_{L^2(\varOmega_r\cup \varOmega_y)}+\left\| \mathcal{P}_N u_{L_0}\right\|_{L^2(\varOmega_r\cup \varOmega_y)}) = :I_1+I_2 \end{align*}

    For I_1 , we have

    \left\|u_{L_0}\right\|_{L^2(\varOmega_r\cup \varOmega_y)}\leq \Big( \int_0^{1-\tau} e^{- 2\beta_1(1-x) / \varepsilon} d x\Big)^{1/2}\leq C\varepsilon^{1/2} N^{-(p+1)}.

    For I_2 , using (4.5) and the decay property (2.3) of u_{L_0} ,

    \begin{align} \begin{aligned} \left\| \mathcal{P}_N u_{L_0}\right\|_{L^2(\Omega_{r} \cup \Omega_{y})} & = \left\{\sum\limits_{i = 1}^{N / 2} \int_{x = x_{i-1}}^{x_i} \int_{y = 0}^1\left[ \mathcal{P}_N u_{L_0}(x, y)\right]^2 d y d x\right\}^{1 / 2} \\ & \leq C\left\{\sum\limits_{i = 1}^{N / 2} \int_{x = x_{i-1}}^{x_i} \int_{y = 0}^1 e^{-2 \beta_1\left(1-x_i\right) / \varepsilon} d y d x\right\}^{1 / 2} \\ & \leq C\left\{\sum\limits_{i = 1}^{N / 2-1} \int_{x = x_i}^{x_{i+1}} \int_{y = 0}^1 e^{-2 \beta_1(1-x) / \varepsilon} d y d x+\int_{x = x_{N / 2-1}}^{x_{N / 2}} \int_{y = 0}^1 e^{-2 \beta_1 \tau / \varepsilon} d y d x\right\}^{1 / 2} \\ & \leq C\left\{\left(\varepsilon+N^{-1}\right) N^{-2(p+1)}\right\}^{1 / 2} , \end{aligned} \end{align} (4.12)

    where we have used that \varphi(1/2)\geq \ln N. Applying Lemma 4.2 and the decay property (2.3), we have for any K\subset\Omega_{x}\cup \Omega_{xy}

    \begin{align*} \left\|u_{L_0}- \mathcal{P}_N u_{L_0}\right\|_{L^2(K)}& \leq C(h_{x, K}^{p+1} \left\|\frac{\partial^{p+1} u_{L_0}}{\partial x^{p+1} }\right\|_{L^2(K)}+h_{y, K}^{p+1} \left\|\frac{\partial^{p+1} u_{L_0}}{\partial y^{p+1}}\right\|_{L^2(K)})\\ &\leq C \Big( (\cfrac{h_{x, K}}{\varepsilon})^{p+1}+h_{y, K}^{p+1}\Big)\Big(\int_{1-\tau}^1\int_0^1 e^{- 2\beta_1(1-x) / \varepsilon}d ydx\Big)^{1/2}\\ &\leq C \Theta_i^{p+1}\leq C (N^{-1} \max \left|\psi^{\prime}\right|)^{p+1}, \end{align*}

    where we have used \cfrac{h_{x, K}}{\varepsilon}\geq CN^{-1}\geq Ch_{y, K} and Lemma 2.2. Similarly, one can show that (4.11) holds for u_{L_1} as well.

    For K\subset \varOmega_r\cup \varOmega_x\cup \varOmega_y , one can prove as above \sum_{K\subset \varOmega_r\cup\varOmega_x\cup \varOmega_y}\left\|u_{L_2}- \mathcal{P}_N u_{L_2}\right\|_{L^2(K)}\leq C N^{-(p+1)} . For K\subset \varOmega_{xy} , we obtain

    \begin{align*} \left\|u_{L_2}- \mathcal{P}_N u_{L_2}\right\|_{L^2(K)}& \leq C(h_{x, K}^{p+1} \left\|\frac{\partial^{p+1} u_{L_2}}{\partial x^{p+1} }\right\|_{L^2(K)}+h_{y, K}^{p+1} \left\|\frac{\partial^{p+1} u_{L_2}}{\partial y^{p+1}}\right\|_{L^2(K)})\\ &\leq C \Big( (\cfrac{h_{x, K}}{\varepsilon})^{p+1}+(\cfrac{h_{y, K}}{\varepsilon})^{p+1}\Big)\Big(\int_{1-\tau}^1\int_{1-\tau_y}^1 e^{- (2\beta_1(1-x)+2\beta_2(1-y)) / \varepsilon} d y dx\Big)^{1/2}\\ &\leq C (\Theta_i^{p+1}+\Theta_j^{p+1})\leq C (N^{-1} \max \left|\psi^{\prime}\right|)^{p+1}. \end{align*}

    Thus, we complete the proof of (4.11). The proof of the lemma is now completed.

    Lemma 4.6. Let u\in H^{p+1}(\varOmega) . Assume that the conditions of Lemma 4.5 hold. Then, we have

    \begin{align*} &\|\nabla(u_{R}- \mathcal{P}_{N}u_{R})\|_{L^2(\Omega)} \leq CN^{-p}, \\ &\|\Delta(u_{R}- \mathcal{P}_{N}u_{R})\|_{L^2(\Omega)} \leq CN^{1-p}, \\ &\Big \Vert \cfrac{\partial^l(u_L- \mathcal{P}_Nu_L)}{\partial x^l}\Big\Vert _{L^2(\varOmega r)} \leq C\varepsilon ^{1/2-l}N^{-(p+1)}, \\ &\Big\Vert \cfrac{\partial^l(u_L- \mathcal{P}_Nu_L)}{\partial x^l}\Big\Vert _{L^2(\varOmega\setminus \varOmega_r)}\leq C\varepsilon ^{1/2-l}(N^{-1} \max \left|\psi^{\prime}\right|)^{p+1-l}, \\ &\Big \Vert \cfrac{\partial^l(u_L- \mathcal{P}_Nu_L)}{\partial y^l}\Big\Vert _{L^2(\varOmega r)} \leq C\varepsilon ^{1/2-l}N^{-(p+1)}, \\ &\Big\Vert \cfrac{\partial^l(u_L- \mathcal{P}_Nu_L)}{\partial y^l}\Big\Vert _{L^2(\varOmega\setminus \varOmega_r)} \leq C\varepsilon ^{1/2-l}(N^{-1} \max \left|\psi^{\prime}\right|)^{p+1-l} \end{align*}

    for l = 1, 2 , where u_L denotes u_{L_0}, \; u_{L_1} , or u_{L_2} .

    Proof. The first and second estimates follow from Lemma 4.3, Lemma 4.4, and the fact that \max\{h_{x, K}, h_{y, k}\}\leq C N^{-1} . From the triangle inequality and (4.8) and (4.10) of Lemma 4.5, we have

    \begin{align*} \Vert \nabla (u_L- \mathcal{P}_N u_L)\Vert_{L^2(\varOmega_r)}\leq \Vert \nabla u_L\Vert_{L^2(\varOmega_r)}+ \Vert \nabla \mathcal{P}_N u_L\Vert_{L^2(\varOmega_r)}\leq C \varepsilon^{-1/2}N^{-(p+1)}, \end{align*}

    where we have used that \varepsilon\leq CN^{-1} . This completes the proof of the third and fifth inequalities for l = 1 . An inverse inequality and (4.9) and (4.10) of Lemma 4.5 lead to

    \begin{align*} \Vert \Delta (u_L- \mathcal{P}_N u_L)\Vert_{L^2(\varOmega_r)}&\leq \Vert \Delta u_L\Vert_{L^2(\varOmega_r)}+ C N \Vert \nabla \mathcal{P}_N u_L\Vert_{L^2(\varOmega_r)}\\&\leq C \varepsilon ^{-3/2}[1+(\varepsilon N)^{3/2}+(\varepsilon N)^{2}]N^{-(p+1)}\\&\leq C\varepsilon ^{-3/2}N^{-(p+1)}, \end{align*}

    where again we have used \varepsilon N\leq C . This proves the third and fifth inequalities for l = 2 . Using Lemma 4.4 with r = p+1-\ell for \ell = 1, 2 and the decay bound (2.3) of u_{L_0} , one can show that for any K\subset \varOmega_x\cup \varOmega_{xy} ,

    \begin{align*} \left\|\cfrac{\partial^\ell (u_{L_0}- \mathcal{P}_N u_{L_0})}{\partial x^\ell}\right\|_{L^2(K)}& \leq C(h_{x, K}^{p+1-\ell} \left\|\frac{\partial^{p+1} u_{L_0}}{\partial x^{p+1} }\right\|_{L^2(K)}+h_{y, K}^{p+1-\ell} \left\|\frac{\partial^{p+1} u_{L_0}}{\partial x^\ell\partial y^{p+1-\ell}}\right\|_{L^2(K)})\\ &\leq C \varepsilon^{-\ell}\Big( (\cfrac{h_{x, K}}{\varepsilon})^{p+1-\ell}+h_{y, K}^{p+1-\ell}\Big)\Big(\int_{1-\tau}^1\int_0^1 e^{- 2\beta_1(1-x) / \varepsilon}d ydx\Big)^{1/2}\\ &\leq C \varepsilon^{1/2-\ell} \Theta_i^{p+1-\ell}\leq C \varepsilon^{1/2-\ell} (N^{-1} \max \left|\psi^{\prime}\right|)^{p+1-\ell}. \end{align*}

    Similarly, one can prove that the result holds for u_{L_1} , too.

    For K\subset \varOmega_r\cup \varOmega_x\cup \varOmega_y , we get \sum_{K\subset \varOmega_r\cup\varOmega_x\cup \varOmega_y}\left\|\cfrac{\partial^\ell (u_{L_2}- \mathcal{P}_N u_{L_2})}{\partial x^\ell}\right\|_{L^2(K)}\leq C \varepsilon^{1/2-\ell}N^{-(p+1)} . For K\subset \varOmega_{xy} , we obtain

    \begin{align*} \left\|\cfrac{\partial^\ell (u_{L_2}- \mathcal{P}_N u_{L_2})}{\partial x^\ell}\right\|_{L^2(K)}& \leq C(h_{x, K}^{p+1-\ell} \left\|\frac{\partial^{p+1} u_{L_2}}{\partial x^{p+1} }\right\|_{L^2(K)}+h_{y, K}^{p+1-\ell} \left\|\frac{\partial^{p+1} u_{L_2}}{\partial x^\ell y^{p+1-\ell}}\right\|_{L^2(K)})\\ &\leq C\varepsilon^{-\ell} \Big( (\cfrac{h_{x, K}}{\varepsilon})^{p+1-\ell}+(\cfrac{h_{y, K}}{\varepsilon})^{p+1-\ell}\Big)\times\\ &\quad\Big(\int_{1-\tau}^1\int_{1-\tau_y}^1 e^{- 2(\beta_1(1-x)+\beta_2(1-y)) / \varepsilon} d y dx\Big)^{1/2}\\ &\leq C \varepsilon^{1/2-\ell}(\Theta_i^{p+1-\ell}+\Theta_j^{p+1-\ell})\leq C \varepsilon^{1/2-\ell} (N^{-1} \max \left|\psi^{\prime}\right|)^{p+1-\ell}, \end{align*}

    which completes the proof of the fourth inequality. Similarly, one can prove that the last inequality holds true. The proof is now finished.

    Unlike the classical numerical methods such as FEM and the SUPG, the proposed WG-FEM does not have Galerkin orthogonality property. This results in some inconsistency errors in the error bounds. We first give a useful error equation in the following lemma.

    Lemma 5.1. [31] Let u solve the problem (1.1). For v_N = \{v_0, v_b\}\in V_N^0 ,

    \begin{align} -\varepsilon\big(\Delta u, v_0\big)& = \varepsilon(\nabla_w ( \mathcal{P}_N u), \nabla_w v_N\big)-E_d(u, v_N), \end{align} (5.1)
    \begin{align} \left(\boldsymbol{ {\bf b}} \cdot \nabla u, v_{0}\right)& = \left(\boldsymbol{ {\bf b}} \cdot \nabla_{w} ( \mathcal{P}_N u), v_{0}\right)-E_{c}\left(u, v_{N}\right) \end{align} (5.2)
    \begin{align} \left(c u, v_{0}\right)& = \left(c( \mathcal{P}_N u), v_{0}\right)-E_{r}\left(u, v_{N}\right) \end{align} (5.3)

    where

    \begin{align} E_d(u, v_N)& = -\varepsilon\big(\nabla(u- \mathcal{P}_N u), \nabla v_0\big) +\varepsilon\big\langle \nabla(u- \mathcal{P}_N u)\cdot {\bf n}, v_0-v_b\big\rangle, \end{align} (5.4)
    \begin{align} E_c(u, v_N)& = \big( u- \mathcal{P}_N u, \nabla \cdot ( {\bf b} v_0)\big)-\big\langle u- \mathcal{P}_N u , {\bf b} \cdot {\bf n} v_0\big\rangle, \end{align} (5.5)
    \begin{align} E_r(u, v_N)& = \big( c( \mathcal{P}_N u)-cu, v_0\big). \end{align} (5.6)

    The following error equation e_N = \{e_0, e_b\}: = \{ \mathcal{P}_N u-u_0, \mathcal{P}_N u-u_b\} will be needed in the error analysis.

    Lemma 5.2. Let u and u_N be the solutions of (1.1) and (2.15), respectively. For v_N = \{v_0, v_b\}\in V_N^0 , one has

    \begin{align} (e_0^{\prime}, v_0)+{\cal A}_{w}(e_N, v_N) = E(u, v_N), \end{align} (5.7)

    where E(u, v_N): = E_d(u, v_N)+E_c(u, v_N)+E_r(u, v_N) , which are defined by (5.4), (5.5), and (5.6), respectively.

    Proof. Multiplying (1.1) by a test function v_N = \{v_0, v_b\}\in V_N^0 , we arrive at

    \begin{align*} (u_t, v_0)-\varepsilon\left(\Delta u, v_{0}\right)+\left( {\bf b}\cdot \nabla u, v_{0}\right)+\left(cu, v_{0}\right) = \left(f, v_{0}\right). \end{align*}

    With the help of (5.1), (5.2), and (5.3), the above equation becomes

    \begin{align*} ( u_t, v_0)+\varepsilon\left(\nabla_{w} \mathcal{P}_N u, \nabla_{w} v_N \right)+\left( {\bf b} \cdot \nabla_{w} \cdot\left( \mathcal{P}_N u\right), v_{0}\right)+\left(c \mathcal{P}_N u, v_{0}\right) = (f, v_0)+E(u, v_N). \end{align*}

    Since \mathcal{P}_N u is continuous in \varOmega , we get

    {\cal S}_c( \mathcal{P}_N u, v_N) = {\cal S}_d( \mathcal{P}_N u, v_N) = 0.

    Therefore, we have

    \begin{equation} ( u_t, v_0)+ {\cal A}_{w}( \mathcal{P}_N u, v_N) = (f, v_0)+E(u, v_N). \end{equation} (5.8)

    Subtracting (2.15) from (5.8) gives (5.7), which completes completed.

    Lemma 5.3. Let \mathcal{P}_N u be the vertex-edge-cell interpolation of the solution u of the problem (1.1). Then, there holds

    \begin{align*} \Big\vert (u- \mathcal{P}_Nu)(x, y)\Big\vert \leq \begin{cases} C N^{-(p+1)}, & \mathit{\text{if}}\ (x, y)\in \varOmega_r, \\ C (N^{-1} \max \left|\psi^{\prime}\right|)^{p+1}, & \mathit{\text{otherwise}}. \end{cases} \end{align*}

    Proof. The solution decomposition (2.1) implies that

    u- \mathcal{P}_N u = \left(u_R- \mathcal{P}_N u_R\right)+\left(u_{L_0}- \mathcal{P}_N u_{L_0}\right)+\left(u_{L_1}- \mathcal{P}_N u_{L_1}\right)+\left(u_{L_2}- \mathcal{P}_N u_{L_2}\right) .

    Using Lemma 4.2 with q = \infty and Lemma 2.1,

    \left|\left(u_R- \mathcal{P}_N u_R\right)(x, y)\right| \leq C N^{-(p+1)}\sum\limits_{i+j = p+1}\|\cfrac{\partial^{p+1}u_R}{\partial x^i \partial y^j}\|_{L^{\infty}(\varOmega)} \leq C N^{-(p+1)} \quad \forall(x, y) \in \varOmega .

    Next, we examine the layer parts one by one. Let (x, y)\in K\subset \varOmega_r\cup\varOmega_y . From the L^\infty stability property (4.5) of the interpolation operator, one has

    \left|\left(u_{L_0}- \mathcal{P}_N u_{L_0}\right)(x, y)\right| \leq C \Vert u_{L_0} \Vert_{L^\infty(K)}\leq C e^{-\beta_1(1-x_i)/\varepsilon}\leq C e^{-\beta_1\tau/\varepsilon} \leq C N^{-(p+1)}.

    Let (x, y)\in K \subset \Omega_{x} \cup \Omega_{xy} . The stability property (4.5) and Lemma 4.2 with q = \infty yield

    \begin{align*} \Vert u_{L_0}- \mathcal{P}_N u_{L_0}\Vert_{L^\infty(K)}& \leq C \min \Big\{ \Vert u_{L_0}\Vert_{L^\infty(K)}, h_{x, K}^{p+1}\Vert\cfrac{\partial^{p+1} u_{L_0}}{\partial x^{p+1}} \Vert_{L^{\infty}(K)} +h_{x, K}^{p+1}\Vert\cfrac{\partial^{p+1} u_{L_0}}{\partial x^{p+1}} \Vert_{L^{\infty}(K)}\Big\}\\ &\leq \min \Big\{ 1, (\cfrac{h_{x, K}}{\varepsilon})^{p+1}+(h_{y, K})^{p+1}\Big\}e^{- \beta_1(1-x_i) / \varepsilon}\\ &\leq C \Theta_i^{p+1}\leq C (N^{-1} \max \left|\psi^{\prime}\right|)^{p+1}. \end{align*}

    Similarly, we can derive the estimates on the other layer components u_{L_1} and u_{L_2} . Combining the above estimates gives the desired conclusion.

    Thus, we complete the proof.

    We recall the following trace inequality. For any u \in H^1(K) , one has

    \begin{align} \|u\|_e^2 \leq C\left(h_K^{-1}\|u\|_{L^2(K)}^2+\|u\|_{L^2(K)}\|\nabla u\|_{L^2(K)}\right) . \end{align} (5.9)

    Lemma 5.4. Let u\in H^{p+1}(\varOmega) and \rho_K be given by (2.13). Assume that the conditions of Lemma 4.5 hold. Then, one has

    \begin{align*} \left\{ \sum _{K\in \mathcal{T}_N}\frac{\varepsilon ^2}{\rho_K}\Vert \nabla(u- \mathcal{P}_Nu)\Vert _{L^2(\partial K)}^2\right\} ^{1/2} \leq C(N^{-1} \max \left|\psi^{\prime}\right|)^{p}. \end{align*}

    Proof. For the sake of simplicity, we use the following notations. Let \zeta_R: = u_R- \mathcal{P}_N u_R and \zeta_L: = u_L- \mathcal{P}_N u_L represent the interpolation errors of the regular and layer components of the solution. Hence, by the triangle inequality, we have

    \begin{align} \sum _{K\in \mathcal{T}_N}\frac{\varepsilon ^2}{\rho_K}\Vert \nabla \zeta \Vert _{L^2(\partial K)}^2 \leq\sum _{K\in \mathcal{T}_N} \frac{\varepsilon ^2}{\rho_K}(\Vert \nabla\zeta _R\Vert _{L^2(\partial K)}^2 +\Vert \nabla\zeta _L\Vert _{L^2(\partial K)}^2). \end{align} (5.10)

    With the help of the trace inequality (5.9), we have

    \begin{align*} \begin{aligned} \Vert \nabla\zeta_R\Vert _{L^2(\partial K)}^2 \leq C(h_K^{-1}\Vert \nabla\zeta_R\Vert _{L^2(K)}^2 +\Vert \nabla\zeta_R\Vert _{L^2(K)}\Vert \Delta\zeta_R\Vert _{L^2(K)}). \end{aligned} \end{align*}

    Now, appealing the definition (2.13) of stabilization parameter \rho_K and Lemma 4.6 gives

    \begin{align} \begin{split} \sum _{K\in \mathcal{T}_N}\frac{\varepsilon ^2}{\rho_K}\Vert \nabla \zeta_R\Vert _{L^2(\partial K)}^2&\leq C \sum _{K\in \mathcal{T}_N}\frac{\varepsilon ^2}{\rho_K} (h_K^{-1}\Vert \nabla \zeta_R\Vert _{L^2(K)}^2 +\Vert \nabla \zeta_R\Vert _{L^2(K)}\Vert \Delta \zeta_R\Vert _{L^2(K)})\\&\leq C(\varepsilon ^2N\Vert \nabla \zeta_R\Vert _{L^2(\varOmega_r)}^2+\varepsilon \Vert \nabla \zeta_R\Vert _{L^2(\varOmega\setminus\varOmega_r)}^2 \\&\quad +\, \varepsilon ^2\Vert \nabla \zeta_R\Vert _{L^2(\varOmega_r)}\Vert \Delta \zeta_R\Vert _{L^2(\varOmega_r)} +\varepsilon ^2 N^{-1}\ln N\Vert \nabla \zeta_R\Vert _{L^2(\varOmega\setminus\varOmega_r)}\Vert \Delta \zeta_R\Vert _{L^2(\varOmega\setminus\varOmega_r)}) \\&\leq C\varepsilon N^{-2p}, \end{split} \end{align} (5.11)

    where we have used that \varepsilon N < 1 .

    Using once again the trace inequality (5.9), we have

    \begin{align*} \begin{aligned} \Vert \nabla\zeta_L\Vert _{L^2(\partial K)}^2 \leq C(h_K^{-1}\Vert \nabla\zeta_L\Vert _{L^2(K)}^2 +\Vert \nabla\zeta_L\Vert _{L^2(K)}\Vert \nabla\zeta_L\Vert _{L^2(K)}). \end{aligned} \end{align*}

    Now, appealing the definition (2.13) of stabilization parameter \rho_K and Lemma 4.6 again reveals that

    \begin{align} \begin{split} \sum _{K\in \mathcal{T}_N}\frac{\varepsilon ^2}{\rho_K}\Vert \nabla \zeta_L\Vert _{L^2(\partial K)}^2&\leq C \sum _{K\in \mathcal{T}_N}\frac{\varepsilon ^2}{\rho_K} (h_K^{-1}\Vert \nabla \zeta_L\Vert _{L^2(K)}^2 +\Vert \nabla \zeta_L\Vert _{L^2(K)}\Vert \Delta \zeta_L\Vert _{L^2(K)})\\&\leq C(\varepsilon ^2N\Vert \nabla \zeta_L\Vert _{L^2(\varOmega_r)}^2+\varepsilon \Vert \nabla \zeta_L\Vert _{L^2(\varOmega\setminus\varOmega_r)}^2 \\&\quad +\, \varepsilon ^2\Vert \nabla \zeta_L\Vert _{L^2(\varOmega_r)}\Vert \Delta \zeta_L\Vert _{L^2(\varOmega_r)} +\varepsilon ^2 N^{-1}\ln N\Vert \nabla \zeta_L\Vert _{L^2(\varOmega\setminus\varOmega_r)}\Vert \Delta \zeta_L\Vert _{L^2(\varOmega\setminus\varOmega_r)}) \\&\leq C[(\varepsilon +N^{-1})N^{-(2p+1)}+(N^{-1} \max \left|\psi^{\prime}\right|)^{2p}]. \end{split} \end{align} (5.12)

    Plugging (5.12) and (5.11) into (5.10) yields

    \begin{align*} \sum _{K\in \mathcal{T}_N}\frac{\varepsilon ^2}{\rho_K}\Vert \nabla \zeta \Vert _{L^2(\partial K)}^2 \leq C[(\varepsilon +N^{-2})N^{-2p}+(N^{-1} \max \left|\psi^{\prime}\right|)^{2p}]. \end{align*}

    Consequently, we have

    \begin{align*} \left\{ \sum _{K\in \mathcal{T}_N}\frac{\varepsilon ^2}{\rho_K}\Vert \nabla(u- \mathcal{P}_Nu)\Vert _{L^2(\partial K)}^2\right\} ^{1/2} \leq C(N^{-1} \max \left|\psi^{\prime}\right|)^{p}, \end{align*}

    which completes the proof.

    Now, we shall prove the error bounds for the consistency errors.

    Lemma 5.5. (A priori bounds) Assume that {\cal T}_{N} is the tensor product mesh as defined in Section 2. Then, for u\in H^{k+1}(\Omega) and v_N\in V_{N}^0, we have

    \begin{eqnarray} |E_d(u, v_N)| &\leq& C (N^{-1} \max \left|\psi^{\prime}\right|)^{p}\|{v_N}\|_\varepsilon , \end{eqnarray} (5.13)
    \begin{eqnarray} |E_c(u, v_N)+E_r(u, v_N)| &\leq& C(N^{-1} \max \left|\psi^{\prime}\right|)^{p}\|{v_N}\|_\varepsilon , \end{eqnarray} (5.14)

    Proof. It follows from the Cauchy-Schwarz and Holder inequalities that

    \begin{align} |E_d(u, v_N)|&\leq\sum\limits_{K \in \mathcal{T}_{N}}\varepsilon\|\nabla(u- \mathcal{P}_N u)\|_{L^2(K)}\|\nabla v_0\|_{L^2(K)} \\&+\sum\limits_{K \in \mathcal{T}_{N}}\varepsilon\|\nabla(u- \mathcal{P}_N u)\|_{L^2(\partial K)}\|v_0-v_b\|_{L^2(\partial K)}, \\ &: = {\cal S}_{1}+{\cal S}_2. \end{align} (5.15)

    Now, it then follows from Lemma 4.6 that

    \begin{eqnarray} {\cal S}_{1} & = &\sum\limits_{K \in \mathcal{T}_{N}}\varepsilon\|\nabla(u- \mathcal{P}_N u)\|_{L^2(K)}\|\nabla v_0\|_{L^2(K)} \\ & = & \sum\limits_{K \in \mathcal{T}_{N}}\varepsilon^{1/2}\|\nabla(u_{}- \mathcal{P}_N u_{} )\|_{L^2(K)}\varepsilon^{1/2}\|\nabla v_0\|_{L^2(K)} \\ &\leq&\Big(\sum\limits_{K \in \mathcal{T}_{N}}\varepsilon^{1/2}\|\nabla(u_{R}- \mathcal{P}_N u_{R} )\|_{L^2(K)}+\sum\limits_{K \subset\varOmega_r }\varepsilon^{1/2}\|\nabla(u_{L}- \mathcal{P}_N u_{L} )\|_{L^2(K)} \\&&+\sum\limits_{K \subset \varOmega\setminus\varOmega_r}\varepsilon^{1/2}\|\nabla(u_{L}- \mathcal{P}_N u_{L} )\|_{L^2(K)}\Big)\|\nabla v_N\|_\varepsilon \\ &\leq& C(\varepsilon^{1/2}N^{-p}+N^{-(p+1)}+ (N^{-1} \max \left|\psi^{\prime}\right|)^{p})\|v_N\|_\varepsilon \\ &\leq &C (N^{-1} \max \left|\psi^{\prime}\right|)^{p}\|v_N\|_\varepsilon . \end{eqnarray} (5.16)

    Next, we consider the term {\cal S}_{2} . From the Cauchy-Schwarz inequality and Lemma 5.4, we have

    \begin{align} \begin{aligned} {\cal S}_2 & = \varepsilon \sum\limits_{K \in \mathcal{T}_{N}}\|\nabla(u- \mathcal{P}_N u)\|_{L^2(\partial K)}\|v_0-v_b\|_{L^2(\partial K)} \\ &\leq \Biggl\{\sum\limits_{K \in \mathcal{T}_{N}}\frac{\varepsilon^2}{\rho_{K}}\|\nabla(u- \mathcal{P}_{N}u)\|^2_{L^2( \partial K)}\Biggr\}^{1/2}\Biggl\{\sum\limits_{K \in \mathcal{T}_{N}}{\rho_{K}}\|v_0-v_b\|^2_{L^2(\partial K)}\Biggr\}^{1/2}\\ &\leq C (N^{-1} \max \left|\psi^{\prime}\right|)^{p}\|v_N\|_\varepsilon . \end{aligned} \end{align} (5.17)

    Combining (5.16) and (5.17), we get

    \begin{eqnarray} |E_d(u, v_N)|\leq C (N^{-1} \max \left|\psi^{\prime}\right|)^{p}\|v_N\|_\varepsilon . \end{eqnarray} (5.18)

    From (5.5) and (5.6) and using \langle u-I_N u, {\bf b}\cdot {\bf n}v_b\rangle = 0 , we arrive at

    \begin{align*} E_c(u, v_N)+E_r(u, v_N)& = (u- \mathcal{P}_N u, \mathbf{b}\cdot \nabla v_0)+\langle u- \mathcal{P}_N u, {\bf b}\cdot {\bf n}(v_0-v_b)\rangle\\&\; \; \; \; \; \; +(u- \mathcal{P}_N u, (\nabla\cdot \mathbf{b}-c) v_0) \\ & = : \mathcal{R}_1+\mathcal{R}_1^b+\mathcal{R}_2. \end{align*}

    Now, the Hölder inequality and Lemma 5.3 lead us to write

    \begin{eqnarray} \mathcal{R}_1&\leq & C\Big(\sum\limits_{K \subset \varOmega_r}\left\|u- \mathcal{P}_N u\right\|_{L^\infty(K)}\left\|\nabla v_0 \right\|_{L^1(K)}+ \sum\limits_{K \subset \varOmega\setminus \varOmega_r}\left\|u- \mathcal{P}_N u\right\|_{L^\infty(K)}\left\|\nabla v_0\right\|_{L^1(K)}\Big) \\ &\leq&C \Big( N^{-(p+1)} \sum\limits_{K \subset \varOmega_r}\left\|\nabla v_0 \right\|_{L^1(K)}+ C (N^{-1} \max \left|\psi^{\prime}\right|)^{p+1}\sum\limits_{K \subset \varOmega \setminus \varOmega_r}\left\|\nabla v_0 \right\|_{L^1(K)}\Big). \end{eqnarray} (5.19)

    The Cauchy Schwartz and inverse inequalities give

    \begin{align} \begin{split} \sum\limits_{K \subset \varOmega_r}\left\|\nabla v_0 \right\|_{L^1(K)}\leq C N \sum\limits_{K \subset \varOmega_r}\left\| v_0 \right\|_{L^1(K)} \leq C N \vert \varOmega_r\vert^{1/2} \big(\sum\limits_{K \subset \varOmega_r}\left\| v_0 \right\|_{L^2(K)}^2\big)^{1/2} \leq C N \Vert v_N\Vert_\varepsilon . \end{split} \end{align} (5.20)

    Appealing the Cauchy Schwartz inequality on \varOmega\setminus\varOmega_r , we have

    \begin{align} \begin{split} \sum\limits_{K \subset \varOmega \setminus \varOmega_r} \Vert \nabla v_0\Vert_{L^1(K)} &\leq \sum\limits_{K \subset \varOmega_x} \Vert \nabla v_0\Vert_{L^1(K)}+\sum\limits_{K \subset \varOmega_y} \Vert \nabla v_0\Vert_{L^1(K)}+\sum\limits_{K \subset \varOmega_{xy}} \Vert \nabla v_0\Vert_{L^1(K)}\\ & \leq \sqrt{ \tau(1-\tau_y)} \Vert \nabla v_0\Vert_{L^2(\varOmega_x)}+\sqrt{\tau_y(1- \tau)} \Vert \nabla v_0\Vert_{L^2(\varOmega_y)}\\ &\qquad + \sqrt{ \tau \tau_y} \Vert \nabla v_0\Vert_{L^2(\varOmega_{xy})}\\ &\leq C \ln ^{1/2} N \sum\limits_{K \subset \varOmega \setminus \varOmega_r} \varepsilon^{1/2} \Vert \nabla v_0\Vert_{L^2(K)} \\ &\leq C ( \ln N)^{1/2} \Vert v_N\Vert_\varepsilon .\end{split} \end{align} (5.21)

    Using the error bounds (5.20) and (5.21) in (5.19), we obtain

    \begin{align} \begin{split} |\mathcal{R}_1|&\leq C[N^{-p} +(N^{-1} \max \left|\psi^{\prime}\right|)(\ln N)^{1/2} (N^{-1} \max \left|\psi^{\prime}\right|)^{p}]\Vert v_N\Vert _\varepsilon \\&\leq C(N^{-1} \max \left|\psi^{\prime}\right|)^{p}\Vert v_N\Vert _\varepsilon , \end{split} \end{align} (5.22)

    where we have used the fact that N^{-1} (\ln N)^{1/2} \max \left|\psi^{\prime}\right| = N^{-1}(\ln N)^{1/2} \left|\psi^{\prime}(0)\right| < C .

    Since u and \mathcal{P}_N u are continuous, we conclude that \Vert u- \mathcal{P}_N u\Vert_{L^\infty(e)}\leq \Vert u- \mathcal{P}_N u\Vert_{L^\infty(K)} for any e\subset \partial K \in \mathcal{T}_N . Then, the Hölder inequality and Lemma 5.3 imply that

    \begin{align*} \vert \mathcal{R}_1^b\vert &\leq \sum\limits_{K \in \mathcal{T}_N}\Vert u- \mathcal{P}_N u\Vert_{L^\infty(\partial K)}\Vert v_0-v_b\Vert_{L^1(\partial K)} \\ &\leq \sum\limits_{K \in \mathcal{T}_N}\Vert u- \mathcal{P}_N u\Vert_{L^\infty(\partial K)} \Vert v_0-v_b\Vert_{L^2(\partial K)}\vert \partial K\vert ^{1/2}\\ &\leq \Big(\sum\limits_{K \in \mathcal{T}_N} \Vert u- \mathcal{P}_N u\Vert_{L^\infty(\partial K)}^2 \vert \partial K\vert \rho_K^{-1}\Big)^{1/2} ( \sum\limits_{K \in \mathcal{T}_N}\rho_K \Vert v_0-v_b\Vert_{L^2(\partial K)}^2)^{1/2}\\ &\leq C\Big(N^{-2p-1}+(N^{-1} \max \left|\psi^{\prime}\right|)^{2(p+1)}(\varepsilon \ln N) \ln N\Big)^{1/2} ( \sum\limits_{K \in \mathcal{T}_N}\rho_K \Vert v_0-v_b\Vert_{L^2(\partial K)}^2)^{1/2}\\ &\leq C N^{-(p+1/2)} (\max \left|\psi^{\prime}\right|)^{p+1}\Vert v_N\Vert_\varepsilon , \end{align*}

    where we have used that \varepsilon \ln N < C and (N^{-1} \max \left|\psi^{\prime}\right|)^{2(p+1)}\ln N \leq N^{-(2p+1)} (\max \left|\psi^{\prime}\right|)^{2(p+1)} .

    Using the Cauchy Schwartz inequality and (4.6) and (4.11) of Lemma 4.5, we obtain

    \begin{align*} \vert \mathcal{R}_2\vert &\leq C \Vert u- \mathcal{P}_N u\Vert \Vert v_0\Vert \\ & \leq C (N^{-1} \max \left|\psi^{\prime}\right|)^{p+1} \Vert v_N\Vert_\varepsilon . \end{align*}

    The proof is completed.

    By letting v_N = e_{N} in (5.7), we obtain

    \begin{eqnarray*} \frac{1}{2}\frac{d}{dt}\|e_{0}(t)\|^2+{\cal A}_{w}(e_{N}, e_{N})\leq |E_d(u, e_{N})|+|E_c(u, e_{N})|+|E_r(u, e_{N})|. \nonumber \end{eqnarray*}

    It then follows from the estimates (5.13) and (5.14), together with Young's inequality and 3.10, that

    \begin{align*} \frac{1}{2}\frac{d}{dt}\|e_{0}(t)\|^2+C \|{e_{N}}\|_{\varepsilon}^2 &\leq C (N^{-1} \max \left|\psi^{\prime}\right|)^{p}\|{e_{N}}\|_{\varepsilon}\\ &\leq C (N^{-1} \max \left|\psi^{\prime}\right|)^{2p}+\cfrac{C}{2} \|{e_{N}}\|_{\varepsilon}^2. \end{align*}

    As a result,

    \begin{align} \frac{1}{2}\frac{d}{dt}\|e_{0}(t)\|^2+C \|{e_{N}}\|_{\varepsilon}^2 &\leq C (N^{-1} \max \left|\psi^{\prime}\right|)^{2p}. \end{align} (5.23)

    Then, by integrating from 0 to t , we have

    \begin{eqnarray*} \|e_{0}(t)\|^2+C\int_{0}^{t}\|{e_{N}}\|_{\varepsilon}^2ds\leq C\Big\{\|e_{0}(0)\|^2+(N^{-1} \max \left|\psi^{\prime}\right|)^{2p} \Big\}. \end{eqnarray*}

    This result is collected in the following theorem.

    Theorem 5.1. (Semi-discrete estimate) Let u\in H^{k+1}(\Omega) be the solution of (1.1)-(1.2) and u_{N}\in V_{N}^0 be the solution of (2.15). Then, we have

    \begin{eqnarray} \|e_{0}(t)\|^2+\int_{0}^{t}\|{e_{N}}\|_{\varepsilon}^2ds\leq C\Big\{\|e_{0}(0)\|^2+(N^{-1} \max \left|\psi^{\prime}\right|)^{2p} \Big\}. \end{eqnarray}

    In this section, we shall use the Crank-Nicolson scheme on uniform time mesh in time to derive the fully discrete approximation of the problem (1.1) and (1.2). For a given partition 0 = t_0 < t_1 < \cdots < t_M = T of the time interval J = [0, T] for some positive integer M and step length \tau = \frac{T}{M} , we define

    \begin{eqnarray*} \partial_\tau \omega^n = \frac{\omega^{n+1}-\omega^n}{\tau}\;\;\;\;\mbox{and}\;\;\;\;\omega^{n+\frac{1}{2}} = \frac{1}{2}(\omega^{n+1}+\omega^n), \;\; 0\leq n\leq M-1, \end{eqnarray*}

    where the sequence \{\omega^n\}_{n = 0}^M \subset L^2({\cal D}) . For simplicity, we denote \xi(\cdot, t_n) by \xi^n for a function \xi:[0, T]\rightarrow L^2(\Omega). We now state our fully discrete weak Galerkin finite element approximation. Find U^n_{N} = \{U_0^n, U_b^n\}\in V_N^0 such that

    \begin{eqnarray} (\partial_\tau U^n_{0}, \phi_0)+{\cal A}_{w}(U^{n+\frac{1}{2}}, \phi_N) = (f^{n+\frac{1}{2}}, \phi_0)\;\;\forall\phi_N\in V_N^0, \end{eqnarray} (6.1)

    with U^0_{N} = { \mathcal{P}}_Nu(0) and f^{n+\frac{1}{2}} = \frac{1}{2}(f^{n+1}+f^{n}) .

    The following lemma shows that the Crank-Nicolson scheme is unconditionally stable in the L^2 norm.

    Lemma 6.1. Let f\in C(0, T; L^2(\Omega)) . Then, we have the following stability estimate for the fully-discrete scheme (6.1):

    \begin{equation} \|U_0^{n}\|\leq C\Big(\|u^0\|+ T\max\limits_{1\leq n\leq M}\|f({t_n})\|\Big), \; n = 0, 1, 2, \ldots, M. \end{equation} (6.2)

    Proof. Choosing v = U_N^{n+1/2} in (6.1), and using the Cauchy-Schwarz inequality, we get

    \begin{equation*} \frac{1}{2\tau}\left( \|U_0^{n+1}\|^{2}-\|U_0^{n}\|^{2}\right)+ C \Vert U_N^{n+\frac{1}{2}}\Vert_\varepsilon ^{2}\leq C\|f^{n+\frac{1}{2}}\| \|U_0^{n+1/2}\|, \end{equation*}

    where we have used that {\cal A}_{w}(U_N^{n+\frac{1}{2}}, U_N^{n+\frac{1}{2}})\geq C \Vert U_N^{n+\frac{1}{2}}\Vert_\varepsilon ^{2} . Using the fact that a^2-b^2 = (a-b)(a+b) and the coercivity of the bilinear form, we have

    \begin{equation*} \frac{1}{2\tau}\left( \|U_0^{n+1}\|-\|U_0^{n}\|\right)\leq (C/2) \|f^{n+\frac{1}{2}}\|, \end{equation*}

    Let 1\leq j \leq M be an integer. We sum the above inequality from n = 1 to n = j :

    \begin{align*} \|U_0^{j}\|&\leq \|U_0^{0}\|+ C\tau \sum\limits_{n = 1}^{j} \|{f}^{n+\frac{1}{2}}\|\\ &\leq C( \|u^{0}\|+ M\tau \max\limits_{1\leq n\leq M}\|f({t_n})\|). \end{align*}

    Recalling that M\tau = T , the result follows. The proof is now completed.

    Next, we shall present the convergence analysis. To begin, we prove the error estimate of the discretization error {\cal P}_N u(t_n)-U^n_{N} . To this end, we need to derive an error equation involving the error e^n_{N}: = U^n_{N}-{\cal P}_{N}u^n .

    We formulate the error equation for e_{N}^n in the following lemma.

    Lemma 6.2. For v_{N} = \{v_0, v_b\} \in V_{N}^0, we have

    \begin{eqnarray} (\partial_{\tau}e_{0}^{n}, v_0)+{\cal A}_{w}(e_{N}^{n+\frac{1}{2}}, v_N)& = & (\xi^n, v_0)+E(u^{n+\frac{1}{2}}, v_N), \end{eqnarray} (6.3)

    where \xi^n = \frac{1}{\tau}(\mathcal{P}_N u^{n+1}- \mathcal{P}_N u^n)-\partial_{t}u^{n+\frac{1}{2}} ; and E(u, v_N) = E_d(u, v_N)+E_c(u, v_N)+E_r(u, v_N).

    Proof. From (1.1), one obtains the following equation:

    \begin{eqnarray} \partial_{t}u^{n+\frac{1}{2}}-\varepsilon\Delta u^{n+\frac{1}{2}}+{ {\bf b}}\cdot \nabla u^{n+\frac{1}{2}}+c u^{n+\frac{1}{2}} = f^{n+\frac{1}{2}}. \end{eqnarray} (6.4)

    On each element T\in {\cal T}_{N}, for v_N = \{v_0, v_b\}\in V_{N}^0, we test equation (6.4) against v_0 to arrive at

    \begin{eqnarray} (f^{n+\frac{1}{2}}, v_0) & = &(\partial_{t}u^{n+\frac{1}{2}}, v_0)_{{\cal T}_{N}} -\sum\limits_{T\in {\cal T}_{N}}(\varepsilon\Delta u^{n+\frac{1}{2}}, v_0)_{T} \\&&+\sum\limits_{T\in {\cal T}_{N}}({\bf b} \cdot\nabla u^{n+\frac{1}{2}}, v_0)_{T}+\big(c u^{n+\frac{1}{2}}, v_0\big)_{{\cal T}_{N}}. \end{eqnarray} (6.5)

    Using a similar argument in deriving (5.8), one can show that

    \begin{eqnarray} (\partial_{t}u^{n+\frac{1}{2}}, v_0)_{{\cal T}_{N}}+{\cal A}_{w}({\cal P}_{N}u^{n+\frac{1}{2}}, v_N) = (f^{n+\frac{1}{2}}, v_0)+E_d(u^{n+\frac{1}{2}}, v_N)+ E_c(u^{n+\frac{1}{2}}, v_N)+E_r(u^{n+\frac{1}{2}}, v_N), \end{eqnarray}

    where we have used that {\cal S}_{d}({\cal P}_{N}u^{n+\frac{1}{2}}, v_N) = 0\; \; \text{ and }\; \; {\cal S}_{c}({\cal P}_{N}u^{n+\frac{1}{2}}, v_N) = 0 since {\cal P}_{N}u is continuous in \Omega . Thus, we get

    \begin{eqnarray} \frac{1}{\tau}({\cal P}_Nu^{n+1}-{\cal P}_Nu^n, v_0)_{{\cal T}_{N}}+{\cal A}_{w}({\cal P}_{N}u, v_N) = (f^{n+\frac{1}{2}}, v_0)+E(u, v_N)+(\xi^n, v_0)_{{\cal T}_{N}}, \end{eqnarray} (6.6)

    Subtracting (6.1) from (6.6) gives the conclusion. We complete the proof.

    Lemma 6.3. Let u\in H^{k+1}(\Omega) . Assume that u and U^n_{N} are the solutions (1.1), (1.2), and (6.1), respectively. One has for n = 1, 2, \ldots, M ,

    \begin{align} \|e_0^{n}\|^2+ C\tau\sum\limits_{j = 0}^{n-1}\|{e_N^{j+\frac{1}{2}}}\|_{\varepsilon}^2\leq C\Big(\tau^4\int_{0}^{t_{n}}\|u_{ttt}(s)\|^2ds+ (N^{-1} \max \left|\psi^{\prime}\right|)^{2p}\Big). \end{align} (6.7)

    Proof. Choosing v_N = e_{N}^{n+\frac{1}{2}} in (6.3) and by the coercivity property (3.6), we find

    \begin{eqnarray} \frac{1}{2\tau}\big(\|e_0^{n+1}\|^2-\|e_0^{n}\|^2\big)+C\|{e_N^{n+\frac{1}{2}}}\|_\varepsilon ^2 &\leq & (\xi^n, e_{0}^{n+\frac{1}{2}}) +E(u^{n+\frac{1}{2}}, e_{N}^{n+\frac{1}{2}}), \end{eqnarray}

    or, equivalently,

    \begin{eqnarray} \|e_0^{n+1}\|^2-\|e_0^{n}\|^2+2 C\tau\|{e_N^{n+\frac{1}{2}}}\|_\varepsilon ^2 &\leq& 2\tau(\xi^n, e_{0}^{n+\frac{1}{2}})+2\tau E(u^{n+\frac{1}{2}}, e_{N}^{n+\frac{1}{2}})\\ &: = & \mathcal{W}_1+\mathcal{W}_2. \end{eqnarray} (6.8)

    We can express the term \xi^n = (\partial_\tau \mathcal{P}_N u^n-\partial_\tau u^n)+ (\partial_\tau u^n-\partial_t u^{n+\frac{1}{2}}) = : T_1+T_2 . We write

    \begin{align} \begin{aligned} T_1& = \cfrac{1}{\tau} \int_{t_n}^{t_{n+1}}\frac{\partial}{\partial t}( \mathcal{P}_N u(\cdot, s)-u(\cdot, s))\, ds\leq \cfrac{1}{\tau} \int_{t_n}^{t_{n+1}} \vert \mathcal{P}_N u_t(\cdot, s) -u_t(\cdot, s)\vert \, ds, \end{aligned} \end{align} (6.9)

    and

    \begin{eqnarray} \begin{aligned} T_2& = \frac{1}{\tau}\Big(u(t_{n+1})-u(t_n)\Big)-\frac{1}{2}\Big(u_t(t_{n+1})+u_t(t_n)\Big)\\ & = \frac{1}{2\tau}\Big(\int_{t_n}^{t_{n+1}}(t_{n+1}-s)(t_n-s)u_{ttt}(s)ds \Big).\end{aligned} \end{eqnarray} (6.10)

    From (6.9) and (6.10), we obtain

    \begin{align} \|\xi^n\|^2&\leq \int_{\Omega}\Bigg[ \frac{1}{2\tau}\Bigg[\int_{t_n}^{t_{n+1}}(t_{n+1}-s)(t_n-s)u_{ttt}(s)ds\Bigg]^2dx \\ &\quad +\int_{\Omega}\Bigg[\cfrac{1}{\tau} \int_{t_n}^{t_{n+1}} \vert \mathcal{P}_N u_t (\cdot, s) - u_t (\cdot, s)\vert \, ds \Bigg]^2dx \\ &\leq \frac{1}{4\tau^2} \int_{\Omega}\Bigg[\int_{t_n}^{t_{n+1}}(t_{n+1}-s)^2(t_n-s)^2ds\int_{t_n}^{t_{n+1}}u_{ttt}^2(s)ds\Bigg]dx\\ &\quad +\cfrac{1}{\tau^2}\int_{\Omega}\Bigg[ \int_{t_n}^{t_{n+1}} \vert \mathcal{P}_N u_t (\cdot, s) - u_t (\cdot, s)\vert^2 \, ds \Bigg]dx \\ &\leq\frac{\tau^3}{120}\int_{t_n}^{t_{n+1}}\|u_{ttt}(s)\|^2dt+ \cfrac{1}{\tau} \Vert \mathcal{P}_N u_t - u_t \Vert_{L^\infty(0, T;L^2(\varOmega))}^2 . \end{align} (6.11)

    Hence, with the aid of the Cauchy-Schwarz and the Poincare inequality, \mathcal{W}_1 in (6.8) can be estimated as follows.

    \begin{eqnarray} |\mathcal{W}_1|& = & |2\tau(\xi^n, e_{0}^{n+\frac{1}{2}})|\leq 2\tau\|\xi^n\|\|e_{0}^{n+\frac{1}{2}}\| \leq2\tau\|\xi^n\|\|{e_{N}^{n+\frac{1}{2}}}\|_\varepsilon \\ &\leq & \frac{\tau}{ C}\|\xi^n\|^2+\tau C\|{e_{N}^{n+\frac{1}{2}}}\|_\varepsilon ^2\\ &\leq & C(\tau^4\int_{t_n}^{t_{n+1}}\|u_{ttt}(s)\|^2ds+\Vert \mathcal{P}_N u_t - u_t \Vert_{L^\infty(0, T;L^2(\varOmega))}^2)+ C\tau\|{e_{N}^{n+\frac{1}{2}}}\|_\varepsilon ^2 \\ &\leq & C(\tau^4\int_{t_n}^{t_{n+1}}\|u_{ttt}(s)\|^2ds+ (N^{-1} \max \left|\psi^{\prime}\right|)^{2p+2}+C\tau\|{e_{N}^{n+\frac{1}{2}}}\|_\varepsilon ^2, \end{eqnarray} (6.12)

    where we have used the Young's inequities in the second inequality, and the estimate (6.11) and Lemma 4.5 in the second estimates of the righthand side. Applying Lemma 5.5 and Young's inequality, we obtain the estimate of the term I_2 in the righthand side of (6.8) as follows:

    \begin{eqnarray} |I_2| &\leq & C \big(2\tau (N^{-1} \max \left|\psi^{\prime}\right|)^{p}\big)\|{e_{N}^{n+\frac{1}{2}}}\|_\varepsilon \\ &\leq & C \tau (N^{-1} \max \left|\psi^{\prime}\right|)^{2p}+ C\tau\|{e_{N}^{n+\frac{1}{2}}}\|_\varepsilon ^2. \end{eqnarray} (6.13)

    Combining (6.8)–(6.13) yields

    \begin{align} &\|e_0^{n+1}\|^2-\|e_0^{n}\|^2+ C\tau\|{e_N^{n+\frac{1}{2}}}\|_\varepsilon ^2\\&\; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \leq C\tau^4\int_{t_n}^{t_{n+1}}\|u_{ttt}(s)\|^2ds+C\tau (N^{-1} \max \left|\psi^{\prime}\right|)^{2p}. \end{align}

    Let 1 \leq j \leq M . Using the fact that e_N^0 = 0 , we sum the above expression from n = 0 to n = j-1 to obtain

    \begin{eqnarray} \|e_0^{j}\|^2+ C\tau\sum\limits_{n = 0}^{j-1}\|{e_N^{n+\frac{1}{2}}}\|_\varepsilon ^2&\leq & C\tau^4\int_{0}^{t_{j}}\|u_{ttt}(s)\|^2ds+C\sum\limits_{n = 0}^{j-1}\tau (N^{-1} \max \left|\psi^{\prime}\right|)^{2p} \\&\leq& C(\tau^4\int_{0}^{t_{j}}\|u_{ttt}(s)\|^2ds+(N^{-1} \max \left|\psi^{\prime}\right|)^{2p}). \end{eqnarray}

    We complete the proof.

    Theorem 6.1. Let u\in H^{k+1}(\Omega) . Assume that u and U^n_{N} are the solutions (1.1), (1.2), and (6.1), respectively. One has for n = 1, 2, \ldots, M ,

    \begin{eqnarray*} \|{e_N^{n+1}}\|_\varepsilon ^2\leq C\big(\tau^4\int_{0}^{t_{m}}\|u_{ttt}(s)\|^2ds+ C(N^{-1} \max \left|\psi^{\prime}\right|)^{2p}\big). \end{eqnarray*}

    Proof. Choosing v = \partial_{\tau}e_N^{n} in (6.3) and by coercivity (3.6), we find

    \begin{eqnarray} \|\partial_{\tau}e_0^{n}\|^2+{\cal A}_{w}({e_N^{n+\frac{1}{2}}}, \partial_{\tau}e_N^{n} )& = & (\xi^n, \partial_{\tau}e_0^{n})+E(u^{n+\frac{1}{2}}, \partial_{\tau}e_N^{n}). \end{eqnarray}

    or, equivalently,

    \begin{align} 2\tau\|\partial_{\tau}e_0^{n}\|^2+{\cal A}_{w}({e_N^{n+1}}, e_N^{n+1} )-{\cal A}_{w}({e_N^{n}}, e_N^{n} )& = 2\tau(\xi^n, \partial_{\tau}e_0^{n})+2\tau E(u^{n+\frac{1}{2}}, \partial_{\tau}e_N^{n})\\ &\leq \tau\Vert \xi^n\Vert^2+ \tau\Vert \partial_{\tau}e_0^{n})\Vert^2+2\tau E(u^{n+\frac{1}{2}}, \partial_{\tau}e_N^{n}). \end{align}

    Thus, we have

    \begin{align*} \tau\|\partial_{\tau}e_0^{n}\|^2+{\cal A}_{w}({e_N^{n+1}}, e_N^{n+1} )-{\cal A}_{w}({e_N^{n}}, e_N^{n} )\leq \tau\Vert \xi^n\Vert^2+ 2\tau E(u^{n+\frac{1}{2}}, \partial_{\tau}e_N^{n}). \end{align*}

    Because e_0^0 = 0 , we sum up the above term from n = 0 to n = m-1 for any fixed m to get

    \begin{align} \sum\limits_{n = 0}^{m-1} \tau\left\|{\partial}_\tau e_{0}^n\right\|^2+C\left\|e_N^m\right\|_\varepsilon^2 \leq\sum\limits_{n = 0}^{m-1} \tau\left\|\xi^n\right\|^2+\sum\limits_{n = 0}^{m-1} 2 \tau E\left(u^{n+1 / 2}, {\partial}_\tau e_N^n\right). \end{align} (6.14)

    From (6.11), we have

    \begin{align} \sum\limits_{n = 0}^{m-1} \tau\left\|\xi^{n}\right\|^2 \leq \frac{\tau^4}{120} \int_0^{t_m}\left\|u_{t t t}(s)\right\|^2 \mathrm{\; d} s +\Vert \mathcal{P}_N u_t - u_t \Vert_{L^\infty(0, T;L^2(\varOmega))}^2. \end{align} (6.15)

    Observe that

    \begin{align} \begin{aligned} & \sum\limits_{n = 0}^{m-1} 2 \tau E\left(u^{n+1 / 2}, {\partial}_\tau e_N^n\right) = \sum\limits_{n = 0}^{m-1} \tau E\left(- {\partial}_\tau u^n, e_N^{n+1 / 2}\right)+2 E\left(u^m, e_N^m\right) \\ & : = J_1+J_2 . \end{aligned} \end{align} (6.16)

    Similar to (6.13), one has

    \begin{align} & \left|J_1\right| \leq C \sum\limits_{n = 0}^{m-1}\tau (N^{-1} \max \left|\psi^{\prime}\right|)^{2p}+ C\tau\sum\limits_{n = 0}^{m-1}\|{e_{N}^{n+\frac{1}{2}}}\|_\varepsilon ^2. \end{align} (6.17)

    It follows from Lemma 5.5, the Cauchy-Schwarz inequality, and Young's inequality that

    \begin{align} \vert J_2\vert \leq C (N^{-1} \max \left|\psi^{\prime}\right|)^{2p}+C\|{e_{N}^m}\|_\varepsilon ^2. \end{align} (6.18)

    From (6.16), (6.17), and (6.18) together with \tau M = T , we have

    \begin{align} \sum\limits_{n = 0}^{m-1} 2 \tau E\left(u^{n+1 / 2}, {\partial}_\tau e_N^n\right)\leq C (N^{-1} \max \left|\psi^{\prime}\right|)^{2p}+C\tau\sum\limits_{n = 0}^{m-1}\|{e_{N}^{n+\frac{1}{2}}}\|_\varepsilon ^2+C\|{e_{N}^m}\|_\varepsilon ^2. \end{align} (6.19)

    Combining (6.14), (6.15), and (6.19) yields that

    \begin{align*} \sum\limits_{n = 0}^{m-1} \tau\left\|{\partial}_\tau e_{0}^n\right\|^2+C\left\|e_N^m\right\|_\varepsilon^2\leq C( \tau^4 \int_0^{t_m}\left\|u_{t t t}(s)\right\|^2 \mathrm{\; d} s+(N^{-1} \max \left|\psi^{\prime}\right|)^{2p}+C\tau\sum\limits_{n = 0}^{m-1}\|{e_{N}^{n+\frac{1}{2}}}\|_\varepsilon ^2). \end{align*}

    Finally, using (6.7), we obtain

    \begin{align*} \sum\limits_{n = 0}^{m-1} \tau\left\|{\partial}_\tau e_{0}^n\right\|^2+C\left\|e_N^m\right\|_\varepsilon^2\leq C( \tau^4 \int_0^{t_m}\left\|u_{t t t}(s)\right\|^2 \mathrm{\; d} s+(N^{-1} \max \left|\psi^{\prime}\right|)^{2p}) \end{align*}

    which completes the proof.

    This section presents various numerical examples for the fully-discrete Crank-Nicolson WG finite element method. We used MATLAB R2020A in our the calculations. We also used the 5-point Gauss-Legendre quadrature rule for evaluating of all integrals. All the calculations were calculated using MATLAB R2016a. The systems of linear equations resulting from the discrete problems were solved by lower-upper (LU) decomposition.

    We apply the fully-discrete WG-FEM on the adaptive meshes shown in Table 1. We choose \sigma = p+1 and calculate the energy-norm \Vert e_N^n\Vert_E and the L^2 -norm error \Vert e_0^n\Vert , where e_N = \{e_0^n, e_b^n\} = \{u-U_0^{n}, u-U_b^n\} is the error using N intervals in each direction. The order of convergence (OC) is computed by the formula

    {O C(2) = \log _2\left(\left\|e_N\right\| /\left\|e_{2 N}\right\|\right) , \quad OC(S) = \cfrac{\log\left(\left\|e_N\right\| /\left\|e_{2 N}\right\|\right)}{\log( 2\log N/\log(2N))}.}

    The numerical errors and the order of convergences in space are also tested. In order for the space error to dominate the errors, we take \tau = N^{-2} for N element in each direction. We list the errors in the energy norm and L^2 -norm and the order of convergence in Tables 2 and 3, respectively. These numerical results show that the order of convergence is of order p and of order p+1 in the energy and L^2 norms, respectively, which support the stated error estimates in Theorem 6.1.

    Table 2.  The energy-error and the order of convergence in space for Example 7.1 \varepsilon = 10^{-5} .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    N \Vert e_N^n\Vert_E OC(S) \Vert e_N^n\Vert_E OC(2) \Vert e_N^n\Vert_E OC(2)
    16 1.256 \times 10^{-1} 6.311 \times 10^{-2} 6.604\times 10^{-2}
    32 8.192\times 10^{-2} 0.90 3.381\times 10^{-2} 0.90 3.466 \times 10^{-2} 0.93
    64 5.102\times 10^{-2} 0.93 1.774\times 10^{-2} 0.93 1.819\times 10^{-2} 0.93
    \mathbb{P}_1 128 3.063\times 10^{-2} 0.95 9.182\times 10^{-3} 0.95 9.547\times 10^{-3} 0.93
    256 1.781\times 10^{-2} 0.97 4.687\times 10^{-3} 0.97 4.873\times 10^{-3} 0.97
    512 1.008\times 10^{-2} 0.99 2.359 \times 10^{-3} 1.00 2.453\times 10^{-3} 0.99
    16 2.406\times 10^{-2} 3.344\times 10^{-3} 3.9238\times 10^{-3}
    32 9.828\times 10^{-3} 1.90 9.603\times 10^{-4} 1.80 1.001 \times 10^{-3} 1.97
    64 3.644\times 10^{-3} 1.94 2.573\times 10^{-4} 1.90 2.626 \times 10^{-4} 1.93
    \mathbb{P}_2 128 1.267\times 10^{-3} 1.96 6.705\times 10^{-5} 1.94 6.891\times 10^{-5} 1.93
    256 4.189\times 10^{-4} 1.97 1.687\times 10^{-5} 1.99 1.758\times 10^{-6} 1.97
    512 1.325\times 10^{-4} 2.00 4.246\times 10^{-6} 1.99 4.425\times 10^{-6} 2.00
    16 4.603\times 10^{-3} 5.866\times 10^{-4} 6.402\times 10^{-4}
    32 1.180 \times 10^{-3} 2.90 8.422\times 10^{-5} 2.80 8.576\times 10^{-5} 2.90
    64 2.633\times 10^{-4} 2.94 1.112\times 10^{-5} 2.92 1.125 \times 10^{-5} 2.93
    \mathbb{P}_3 128 5.299\times 10^{-5} 2.97 1.419\times 10^{-6} 2.97 1.435\times 10^{-6} 2.97
    256 1.001\times 10^{-5} 2.98 1.798\times 10^{-7} 2.98 1.806\times 10^{-7} 2.99
    512 1.782\times 10^{-6} 3.00 2.263\times 10^{-8} 3.00 2.273\times 10^{-8} 3.00

     | Show Table
    DownLoad: CSV
    Table 3.  The L^2 error and the order of convergence in space for Example 7.1 \varepsilon = 10^{-5} .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    N \Vert e_0^n\Vert OC(2) \Vert e_0^n\Vert OC(2) \Vert e_0^n\Vert OC(2)
    16 1.045\times 10^{-2} 1.021\times 10^{-2} 1.023\times 10^{-2}
    32 2.723\times 10^{-3} 1.94 2.671\times 10^{-3} 1.94 2.675 \times 10^{-3} 1.94
    64 6.950\times 10^{-4} 1.97 6.944\times 10^{-4} 1.97 6.946 \times 10^{-4} 1.97
    \mathbb{P}_1 128 1.761\times 10^{-5} 1.98 1.760\times 10^{-5} 1.98 1.760\times 10^{-5} 1.98
    256 4.433\times 10^{-6} 1.99 4.430\times 10^{-6} 1.99 4.432 \times 10^{-6} 1.99
    512 1.109\times 10^{-7} 1.99 1.105\times 10^{-7} 1.99 1.108\times 10^{-7} 1.99
    16 2.297\times 10^{-4} 2.286\times 10^{-4} 2.297\times 10^{-4}
    32 3.013\times 10^{-5} 2.93 3.010\times 10^{-5} 2.93 3.011\times 10^{-5} 2.93
    64 3.845\times 10^{-6} 2.97 3.832\times 10^{-6} 2.97 3.844\times 10^{-6} 2.97
    \mathbb{P}_2 128 4.873\times 10^{-7} 2.98 4.862\times 10^{-7} 2.98 4.868\times 10^{-7} 2.98
    256 6.133\times 10^{-8} 2.99 6.130\times 10^{-8} 2.99 6.132\times 10^{-8} 2.99
    512 7.666\times 10^{-9} 3.00 7.662\times 10^{-9} 3.00 7.664\times 10^{-9} 3.00
    16 3.783\times 10^{-5} 3.780\times 10^{-5} 3.782\times 10^{-5}
    32 2.481\times 10^{-6} 3.93 2.479\times 10^{-6} 3.93 2.480 \times 10^{-6} 3.93
    64 1.616\times 10^{-7} 3.94 1.613\times 10^{-7} 3.94 1.614 \times 10^{-7} 3.94
    \mathbb{P}_3 128 1.031\times 10^{-8} 3.97 1.028\times 10^{-8} 3.97 1.030\times 10^{-8} 3.97
    256 6.488\times 10^{-10} 3.99 6.485\times 10^{-10} 3.99 6.487\times10^{-10} 3.99
    512 4.083\times 10^{-11} 4.00 4.077\times 10^{-11} 4.00 4.079\times 10^{-11} 4.00

     | Show Table
    DownLoad: CSV

    Example 7.1. Let {\bf b} = (1, 1) and T = 1 in the problem (1.1). We choose f and u^0 such that the exact solution is

    u(x, y, t) = e^{-t} xy(1-x)(1-y)\kappa(x)\kappa(y),

    where \kappa(z) = 1-e^{-(1-z)/\varepsilon} .

    In Figure 2, we plot the numerical solutions of the WG-FEM using the \mathbb{P}_1 element on the three layer-adapted meshes given in Figure 1 for \varepsilon = 10^{-5} and N = 32 .

    Figure 2.  Numerical solution of Example 7.1 for \varepsilon = 10^{-5} using \mathbb{P}_1 .

    We next present the temporal convergence rate for Example 7.1. In order for the temporal error to dominate the error, we take N = 256 and \varepsilon = 10^{-5} , and use the \mathbb{P}_3 element. We report the results in the L^2 -norm and the energy norm in Tables 4 and 5, respectively. We see that the order of convergence in time is of order \mathcal{O}(\tau^2) , which verifies the theoretical estimate claimed in Theorem 6.1.

    Table 4.  The L^2 error and the order of convergence in time for Example 7.1 \varepsilon = 10^{-5} .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    \tau \Vert e_0^n\Vert OC(2) \Vert e_0^n\Vert OC(2) \Vert e_0^n\Vert OC(2)
    1/2 7.425\times 10^{-3} 7.424\times 10^{-3} 7.425\times 10^{-3}
    1/4 1.818\times 10^{-3} 2.03 1.816\times 10^{-3} 2.03 1.818 \times 10^{-3} 2.03
    1/8 4.545\times 10^{-4} 1.99 4.545\times 10^{-4} 1.99 4.545 \times 10^{-4} 1.99
    1/16 1.136\times 10^{-4} 2.00 1.135\times 10^{-4} 2.00 1.136\times 10^{-4} 2.00

     | Show Table
    DownLoad: CSV
    Table 5.  The energy error and the order of convergence in time for Example 7.1 \varepsilon = 10^{-5} .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    \tau \Vert e_N^n\Vert_E OC(2) \Vert e_N^n\Vert_E OC(2) \Vert e_N^n\Vert_E OC(2)
    1/2 7.425\times 10^{-3} 7.424\times 10^{-3} 7.425\times 10^{-3}
    1/4 1.856\times 10^{-3} 2.00 1.852\times 10^{-3} 2.00 1.855 \times 10^{-3} 2.00
    1/8 4.672\times 10^{-4} 1.99 4.668\times 10^{-4} 1.99 4.669 \times 10^{-4} 1.99
    1/16 1.168\times 10^{-4} 2.00 1.104\times 10^{-4} 2.00 1.106\times 10^{-4} 2.00

     | Show Table
    DownLoad: CSV

    Lastly, we test the robustness of the WG-FEM method with respect to the small parameter \varepsilon for Example 7.1. We take N = 256 and use the \mathbb{P}_1 element for the values of \varepsilon = 10^{-r}, \; r = 5, 6, \dots, 10 . The results are reported in Table 6. These results show that the WG-FEM is robust with respect to the perturbation parameter \varepsilon .

    Table 6.  The energy-error in space for Example 7.1 for the values of \varepsilon .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    \varepsilon \Vert e_N^n\Vert_E \Vert e_N^n\Vert_E \Vert e_N^n\Vert_E
    10^{-5} 1.781\times 10^{-2} 4.687\times 10^{-3} 4.873\times 10^{-3}
    10^{-6} 1.780\times 10^{-2} 4.685\times 10^{-3} 4.877\times 10^{-3}
    10^{-7} 1.780\times 10^{-2} 4.685\times 10^{-3} 4.877\times 10^{-3}
    10^{-8} 1.780\times 10^{-2} 4.685\times 10^{-3} 4.877\times 10^{-3}
    10^{-9} 1.780\times 10^{-2} 4.685\times 10^{-3} 4.686\times 10^{-3}
    10^{-10} 1.780\times 10^{-2} 4.685\times 10^{-3} 4.879\times 10^{-3}

     | Show Table
    DownLoad: CSV

    The order of convergence via loglog plot in the energy norm and L^2 norm are plotted in Figures 3 and 4, respectively, for Example 7.2. We observe that the order of convergence of order p and of order p+1 in the energy and L^2 norms, respectively, which support the stated error estimates in Theorem 6.1 as in Example 7.1. To test the temporal error, we choose N = 256 and \varepsilon = 10^{-5} , and use the \mathbb{P}_1 element. We present the results in the L^2 -norm and the energy norm in Tables 7 and 8, respectively. We see that the order of convergence in time is of order \mathcal{O}(\tau^2) as claimed in Theorem 6.1.

    Figure 3.  The order of convergence in energy norm via loglog plot for Example 7.2 for \varepsilon = 10^{-5} using \mathbb{P}_1 and \mathbb{P}_2 elements.
    Figure 4.  The order of convergence in L^2 -norm via loglog plot for Example 7.2 for \varepsilon = 10^{-5} using \mathbb{P}_1 and \mathbb{P}_2 elements.
    Table 7.  The L^2 error and the order of convergence in time for Example 7.2 \varepsilon = 10^{-5} .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    \tau \Vert e_0^n\Vert OC(2) \Vert e_0^n\Vert OC(2) \Vert e_0^n\Vert OC(2)
    1/2 7.391\times 10^{-2} 7.390\times 10^{-2} 7.391\times 10^{-2}
    1/4 3.217\times 10^{-2} 1.20 3.215\times 10^{-2} 1.20 3.216 \times 10^{-2} 1.20
    1/8 9.901\times 10^{-3} 1.70 9.895\times 10^{-3} 1.70 9.900 \times 10^{-3} 1.70
    1/16 6.449\times10^{-4} 2.00 6.445\times10^{-4} 2.00 6.448\times10^{-4} 2.00

     | Show Table
    DownLoad: CSV
    Table 8.  The energy error and the order of convergence in time for Example 7.2 \varepsilon = 10^{-5} .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    \tau \Vert e_N^n\Vert_E OC(2) \Vert e_N^n\Vert_E OC(2) \Vert e_N^n\Vert_E OC(2)
    1/2 6.521\times 10^{-2} 6.518\times 10^{-2} 6.520\times 10^{-2}
    1/4 1.723\times 10^{-2} 1.92 1.720\times 10^{-2} 1.92 1.722 \times 10^{-2} 1.92
    1/8 4.277\times 10^{-3} 2.01 4.274\times 10^{-3} 2.01 4.276 \times 10^{-3} 2.01
    1/16 1.061\times 10^{-3} 2.02 1.057\times 10^{-3} 2.02 1.060\times 10^{-3} 2.02

     | Show Table
    DownLoad: CSV

    Example 7.2. Let {\bf b} = (1+x, 2-y) , c = (1+x^2+y^2) , and T = 1 in the problem (1.1). We take f and u^0 such that the exact solution is

    u(x, y, t) = e^{t} xy(1-e^{-(3-2x-x^2)/(2\varepsilon)})(1-e^{-(3-4y+y^2)/(2\varepsilon)}).

    We also test the WG-FEM for Example 7.2 for the robustness against \varepsilon . The results are presented in Table 9 for N = 256 and the \mathbb{P}_1 element for the values of \varepsilon = 10^{-r}, \; r = 5, 6, \dots, 10 . Again, one sees that the WG-FEM is the parameter-uniform method.

    Table 9.  The energy-error in space for Example 7.2 for the values of \varepsilon .
    Shishkin Bakhvalov- Shishkin Bakhvalov-type
    \varepsilon \Vert e_N^n\Vert_E \Vert e_N^n\Vert_E \Vert e_N^n\Vert_E
    10^{-5} 7.856\times 10^{-4} 1.432\times 10^{-4} 1.441 \times 10^{-4}
    10^{-6} 7.852\times 10^{-4} 1.430\times 10^{-4} 1.440 \times 10^{-4}
    10^{-7} 7.852\times 10^{-4} 1.430\times 10^{-4} 1.440 \times 10^{-4}
    10^{-8} 7.852\times 10^{-4} 1.430\times 10^{-4} 1.440 \times 10^{-4}
    10^{-9} 7.852\times 10^{-4} 1.430\times 10^{-4} 1.440 \times 10^{-4}
    10^{-10} 7.843\times 10^{-4} 1.439\times 10^{-4} 1.440 \times 10^{-4}

     | Show Table
    DownLoad: CSV

    In this paper, we present the Crack-Nicolson- WG-FEM applied to the singularly perturbed parabolic convection-dominated problems in 2D. We use the Crack-Nicolson scheme in time on uniform mesh and the WG-FEM in space on three layer-adapted meshes: Shishkin, Bakhvalov-Shishkin, and Bakhvalov meshes. We prove (almost) uniform error estimates of order p in the energy norm and second order estimate in time. With the use of a special interpolation operator, the error analysis of the semi-discrete WG-FEM and the fully discrete WG-FEM have been carried out. Various numerical examples are conducted to validate the convergence rate of the proposed method.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors declare there is no conflicts of interest.



    [1] R. L. Bagley, P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheol., 27 (1983), 201–210. https://doi.org/10.1122/1.549724 doi: 10.1122/1.549724
    [2] D. Baleanu, K. Diethelm, E. Scalas, J. J. Trujillo, Fractional calculus: models and numerical methods, World Scientific, 2012. https://doi.org/10.1142/8180
    [3] J. Sabatier, O. P. Agrawal, J. A. T. Machado, Advances in fractional calculus, Springer, 2007. https://doi.org/10.1007/978-1-4020-6042-7
    [4] A. Vinodkumar, M. Gowrisankar, P. Mohankumar, Existence and stability results on nonlinear delay integro-differential equations with random impulses, Kyungpook Math. J., 56 (2016), 431–450. https://doi.org/10.5666/KMJ.2016.56.2.431 doi: 10.5666/KMJ.2016.56.2.431
    [5] G. Pepe, E. Paifelman, A. Carcaterra, Feedback Volterra control of integro-differential equations, Int. J. Control, 96 (2023), 2651–2670. https://doi.org/10.1080/00207179.2022.2109513 doi: 10.1080/00207179.2022.2109513
    [6] H. Jaradat, F. Awawdeh, E. A. Rawashdeh, Analytic solution of fractional integro-differential equations, Ann. Univ. Craiova, 38 (2011), 389. https://doi.org/10.52846/ami.v38i1.389 doi: 10.52846/ami.v38i1.389
    [7] S. A. Murad, R. W. Ibrahim, S. B. Hadid, Existence and uniqueness for solution of differential equation with mixture of integer and fractional derivative, Pak. Acad. Sci., 49 (2012), 33–37.
    [8] X. Liu, M. Jia, B. Wu, Existence and uniqueness of solution for fractional differential equations with integral boundary conditions, Electron. J. Qual. Theory Differ. Equations, 69 (2009), 69. https://doi.org/10.14232/ejqtde.2009.1.69 doi: 10.14232/ejqtde.2009.1.69
    [9] S. A. Murad, A. S. Rafeeq, Existence of solutions of integro-fractional differential equation when \alpha \in(2, 3] through fixed-point theorem, J. Math. Comput. Sci., 11 (2021), 6392–6402. https://doi.org/10.28919/jmcs/6272 doi: 10.28919/jmcs/6272
    [10] J. G. Abulahad, S. A. Murad, Existence, uniqueness and stability theorems for certain functional fractional initial value problem, Al-Rafidain J. Comput. Sci. Math., 8 (2011), 59–70. https://doi.org/10.33899/csmj.2011.163608 doi: 10.33899/csmj.2011.163608
    [11] M. Benchohra, S. Hamani, S. K. Ntouyas, Boundary value problems for differential equations with fractional order, Surv. Math. Appl., 3 (2008), 1–12.
    [12] M. S. Kumar, M. Deepa, J. Kavitha, V. Sadhasivam, Existence theory of fractional order three-dimensional differential system at resonance, Math. Modell. Control, 3 (2023), 127–138. https://doi.org/10.3934/mmc.2023012 doi: 10.3934/mmc.2023012
    [13] S. Zhang, Y. Liu, Existence of solutions for a class of fractional dynamical systems with two damping terms in Banach space, Math. Modell. Control, 2 (2023), 168–180. https://doi.org/10.3934/mmc.2023015 doi: 10.3934/mmc.2023015
    [14] S. A. Murad, H. J. Zekri, S. Hadid, Existence and uniqueness theorem of fractional mixed Volterra-Fredholm integrodifferential equation with integral boundary conditions, Int. J. Differ. Equations, 2011 (2011), 304570. https://doi.org/10.1155/2011/304570 doi: 10.1155/2011/304570
    [15] S. A. Murad, S. B. Hadid, Existence and uniqueness theorem for fractional differential equation with integral boundary condition, J. Fract. Calc. Appl., 3 (2012), 1–9.
    [16] S. Arshad, V. Lupulescu, D. O'Regan, L^p-solutions for fractional integral equations, Fract. Calc. Appl. Anal., 17 (2014), 259–276. https://doi.org/10.2478/s13540-014-0166-4 doi: 10.2478/s13540-014-0166-4
    [17] M. I. Abbas, Existence and uniqueness of solution for a boundary value problem of fractional order involving two Caputo's fractional derivatives, Adv. Differ. Equations, 2015 (2015), 252. https://doi.org/10.1186/s13662-015-0581-9 doi: 10.1186/s13662-015-0581-9
    [18] A. M. A. El-Sayed, S. A. A. El-Salam, L^p-solution of weighted Cauchy-type problem of a diffreintegral functional equation, Int. J. Nonlinear Sci., 5 (2008), 281–288. https://doi.org/10.14232/ejqtde.2007.1.30 doi: 10.14232/ejqtde.2007.1.30
    [19] T. A. Barton, B. Zhang, L^p-solutions of fractional differential equations, Nonlinear Stud., 19 (2012), 161–177.
    [20] A. Refice, M. Inc, M. S. Hashemi, M. S. Souid, Boundary value problem of Riemann-Liouville fractional differential equations in the variable exponent Lebesgue spaces L^{p(.)}, J. Geom. Phys., 178 (2022), 104554. https://doi.org/10.1016/j.geomphys.2022.104554 doi: 10.1016/j.geomphys.2022.104554
    [21] R. P. Agarwal, A. V. Lupulescu, D. O'regan, L^p-Solution for a class of fractional integral equations, J. Integral Equations Appl., 29 (2017), 251–270. https://doi.org/10.1216/jie-2017-29-2-251 doi: 10.1216/jie-2017-29-2-251
    [22] M. A. Almalahi, S. K. Panchal, F. Jarad, T. Abdeljawad, Ulam-Hyers-Mittag-Leffler stability for tripled system of weighted fractional operator with TIME delay, Adv. Differ. Equations, 2021 (2021), 299. https://doi.org/10.1186/s13662-021-03455-0 doi: 10.1186/s13662-021-03455-0
    [23] S. A. Murad, Certain analysis of solution for the nonlinear Two-point boundary-value problem with Caputo fractional derivative, J. Funct. Spaces, 2022 (2022), 1385355. https://doi.org/10.1155/2022/1385355 doi: 10.1155/2022/1385355
    [24] S. A. Murad, Z. A. Ameen, Existence and Ulam stability for fractional differential equations of mixed Caputo-Riemann derivatives, AIMS Math., 7 (2022), 6404–6419. https://doi.org/ 10.3934/math.2022357 doi: 10.3934/math.2022357
    [25] H. Vu, N. V. Hoa, Ulam-Hyers stability for a nonlinear Volterra integro-differential equation, Hacettepe J. Math. Stat., 49 (2020), 1261–1269. https://doi.org/10.15672/hujms.483606 doi: 10.15672/hujms.483606
    [26] B. Liu, Uncertainty theory, Springer-Verlag, 2015. https://doi.org/10.1007/978-3-662-44354-5
    [27] B. Liu, Uncertainty theory: a branch of mathematics for modeling human uncertainty, Springer-Verlag, 2010. https://doi.org/10.1007/978-3-642-13959-8
    [28] B. Liu, Fuzzy process, hybrid process and uncertain process, J. Uncertain Syst., 2 (2008), 3–16.
    [29] A. Berhail, N. Tabouche, Existence and uniqueness of solution for Hadamard fractional differential equations on an infinite interval with integral boundary value conditions, Appl. Math. E-Notes, 20 (2020), 55–69.
    [30] S. A. Murad, A. S. Rafeeq, T. Abdeljawad, Caputo-Hadamard fractional boundary-value problems in \mathfrak{L}^p-spaces, AIMS Math., 9 (2024), 17464–17488. https://doi.org/10.3934/math.2024849 doi: 10.3934/math.2024849
    [31] Y. Arioua, N. Benhamidouche, Boundary value problem for Caputo-Hadamard fractional differential equations, Surv. Math. Appl., 12 (2017), 103–115.
    [32] A. K. Anwara, S. A. Murada, Existence and Ulam stability of solutions for Caputo-Hadamard fractional differential equations, Gen. Lett. Math., 12 (2022), 85–95. https://doi.org/10.31559/glm2022.12.2.5 doi: 10.31559/glm2022.12.2.5
    [33] A. Boutiara, M. Benbachir, K. Guerbati, Boundary value problem for nonlinear Caputo-Hadamard fractional differential equation with Hadamard fractional integral and anti-periodic conditions, Facta Univ. Ser., 36 (2021), 735–748. https://doi.org/10.22190/fumi191022054B doi: 10.22190/fumi191022054B
    [34] A. Lachouri, A. Ardjouni, A. Djoudi, Existence and uniquness of mild solutions of boundary value problem for Caputo-Hadamard fractional differential equations with integral and anti-periodic conditions, J. Fract. Calc. Appl., 12 (2021), 60–68.
    [35] S. N. Rao, A. H. Msmali, M. Singh, A. A. Ahmadin, Existence and uniqueness for a system of Caputo-Hadamard fractional differential equations with multipoint boundary conditions, J. Funct. Spaces, 2020 (2020), 1–10. https://doi.org/10.1155/2020/8821471 doi: 10.1155/2020/8821471
    [36] J. Wang, Y. Zhou, M. Medved, Existence and stability of fractional differential equations with Hadamard derivative, Topol. Methods Nonlinear Anal., 41 (2013), 113–133.
    [37] S. Muthaiah, M. Murugesan, T. N. Gopal, Existence of solutions for nonlocal boundary value problem of Hadamard fractional differential equations, Adv. Theory Nonlinear Anal. Appl., 3 (2019), 579701. https://doi.org/10.31197/atnaa.579701 doi: 10.31197/atnaa.579701
    [38] A. Boutiara, K. Guerbati, M. Benbachir, Caputo-Hadamard fractional differential equation with three-point boundary conditions in Banach spaces, AIMS Math., 5 (2020), 259–272. https://doi.org/10.3934/math.2020017 doi: 10.3934/math.2020017
    [39] G. Adomian, Review of the decomposition method in applied mathematics, J. Math. Anal. Appl., 135 (1988), 501–544. https://doi.org/10.1016/0022-247x(88)90170-9 doi: 10.1016/0022-247x(88)90170-9
    [40] S. A. Murad S. M. Rasheed, Application of Adomian decomposition method for solving fractional differential equation, J. Educ. Sci., 22 (2009), 93–103. https://doi.org/10.33899/edusj.2009.57439 doi: 10.33899/edusj.2009.57439
    [41] P. Guo, The Adomian decomposition method for a type of fractional differential equations, J. Appl. Math. Phys., 7 (2019), 2459–2466. https://doi.org/10.4236/jamp.2019.710166 doi: 10.4236/jamp.2019.710166
    [42] J. G. Abdulahad, S. A. Murad, Local existence theorem of fractional differential equations in Lp space, AL-Rafidain J. Comput. Sci. Math., 9 (2012), 71–78. https://doi.org/10.33899/csmj.2012.163702 doi: 10.33899/csmj.2012.163702
    [43] W. Benhamida, S. Hamani, J. Henderson, Boundary value problems for Caputo-Hadamard fractional differential equations, Adv. Theory Nonlinear Anal. Appl., 2 (2018), 138–145. https://doi.org/10.31197/atnaa.419517 doi: 10.31197/atnaa.419517
    [44] G. Wang, S. Liu, R. P. Agrawal, L. Zhang, Positive solution of integral boundary value problem involving Riemann-Liouville fractional derivative, J. Fract. Calc. Appl., 4 (2023), 312–321. https://doi.org/10.21608/jfca.2023.283809 doi: 10.21608/jfca.2023.283809
    [45] A. A. Kilbas, H. M. Srivastava, J. J. Trujillo, Theory and applications of fractional differential equations, Elsevier, 2006. https://doi.org/10.1016/s0304-0208(06)80001-0
    [46] I. A. Rus, Ulam stabilities of ordinary differential equations in a Banach space, Carpathian J. Math., 26 (2010), 103–107.
    [47] M. A. Krasnosel'skii, Two remarks on the method of successive approximations, Uspekhi Mat. Nauk, 10 (1995), 123–127.
    [48] J. Mikusiński, The Bochner integral, Springer, 1978. https://doi.org/10.1007/978-3-0348-5567-9_3
    [49] A. Granas, J. Dugundji, Fixed point theory, Springer, 2003. https://doi.org/10.1007/978-0-387-21593-8
    [50] H. L. Royden, Real analysis, Prentice-Hall of India Private Limited, 1965.
    [51] F. A. Hendi, W. Shammakh, H. Al-badrani, Existence result and approximate solutions for quadratic integro-differential equations of fractional order, J. King Saud Univ., 31 (2019), 314–321. https://doi.org/10.1016/j.jksus.2018.05.008 doi: 10.1016/j.jksus.2018.05.008
    [52] J. S. Duan, R. Rach, D. Baleanu, A. M. Wazwaz, A review of the Adomian decomposition method and its applications to fractional differential equations, Commun. Fract. Calc., 3 (2012), 73–99.
  • Reader Comments
  • © 2024 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
通讯作者: 陈斌, bchen63@163.com
  • 1. 

    沈阳化工大学材料科学与工程学院 沈阳 110142

  1. 本站搜索
  2. 百度学术搜索
  3. 万方数据库搜索
  4. CNKI搜索

Metrics

Article views(705) PDF downloads(55) Cited by(0)

Figures and Tables

Figures(3)  /  Tables(9)

Other Articles By Authors

/

DownLoad:  Full-Size Img  PowerPoint
Return
Return

Catalog