
Citation: Jolanta Kwiatkowska-Malina, Andrzej Szymon Borkowski. Geostatistical modelling of soil contamination with arsenic, cadmium, lead, and nickel: the Silesian voivodeship, Poland case study[J]. AIMS Geosciences, 2020, 6(2): 135-148. doi: 10.3934/geosci.2020009
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For simplicity, we consider Poisson equation with a Dirichlet boundary condition as our model problem.
−Δu=f,inΩ, | (1) |
u=g,on∂Ω, | (2) |
where
Using integration by parts, we can get the variational form: find
(∇u,∇v)=(f,v),∀v∈H10(Ω). | (3) |
Various finite element methods have been introduced to solve the Poisson equations (1)-(2), such as the Galerkin finite element methods (FEMs)[2, 3], the mixed FEMs [15] and the finite volume methods (FVMs) [6], etc. The FVMs emphasis on the local conservation property and discretize equations by asking the solution satisfying the flux conservation on a dual mesh consisting of control volumes. The mixed FEMs is another category method that based on the variable
The classical conforming finite element method obtains numerical approximate results by constructing a finite-dimensional subspace of
(∇uh,∇vh)=(f,vh),∀vh∈V0h, | (4) |
where
One obvious disadvantage of discontinuous finite element methods is their rather complex formulations which are often necessary to ensure connections of discontinuous solutions across element boundaries. For example, the IPDG methods add parameter depending interior penalty terms. Besides additional programming complexity, one often has difficulties in finding optimal values for the penalty parameters and corresponding efficient solvers. Most recently, Zhang and Ye [21] developed a discontinuous finite element method that has an ultra simple weak formulation on triangular/tetrahedal meshes. The corresponding numerical scheme can be written as: find
(∇wuh,∇wvh)=(f,vh),∀vh∈V0h, | (5) |
where
Following the work in [21, 22], we propose a new conforming DG finite element method on rectangular partitions in this work. It can be obtained from the conforming formulation simply by replacing
In this paper, we keep the same finite element space as DG method, replace the boundary function with the average of the inner function, and use the weak gradient arising from local Raviart-Thomas (RT) elements [5] to approximate the classic gradient. Moreover, the derivation process in this paper is based on rectangular RT elements [16]. Error estimates of optimal order are established for the corresponding conforming DG approximation in both a discrete
The rest of this paper is organized as follows: In Section 2, we shall present the conforming DG finite element scheme for the Poisson equation on rectangular partitions. Section 3 is devoted to a discussion of the stability and solvability of the new method. In Section 4, we shall prepare ourselves for error estimates by deriving some identities. Error estimates of optimal order in
Throughout this paper, we adopt the standard definition of Sobolev space
H10(Ω)={v∈H1(Ω):v|∂Ω=0}, |
and the space
H(div,Ω)={q∈[L2(Ω)]d:∇⋅q∈L2(Ω)}. |
Assume that the domain
For any interior edge
\begin{eqnarray} \{v\}& = &\frac{1}{2}(v|_{\partial T_1}+v|_{\partial T_2}),\; \; [\![ v]\! ] = v|_{\partial T_1}\boldsymbol{n}_1+v|_{\partial T_2}\boldsymbol{n}_2, \end{eqnarray} | (6) |
where
\begin{eqnarray} \{v\}& = &v|_e \; \; \text{and}\; \; [\![ v]\! ] = v|_e\boldsymbol{n}. \end{eqnarray} | (7) |
We define a discontinuous finite element space
\begin{eqnarray} V_h = \{v\in L^2(\Omega): v|_T \in Q_k(T), \; \forall \; T\in \mathcal{T}_h\}, \end{eqnarray} | (8) |
and its subspace
\begin{eqnarray} V^0_h = \{v \in V_h:v = 0 \; \text{on} \; \partial \Omega\}, \end{eqnarray} | (9) |
where
Definition 2.1. For a given
\begin{eqnarray} (\nabla_{d}v,\boldsymbol{q})_T: = -(v,\nabla\cdot\boldsymbol{q})_T +\langle \{v\},\boldsymbol{q}\cdot \boldsymbol{n}\rangle_{\partial T},\; \; \; \; \forall\; \boldsymbol{q}\in RT_k(T), \end{eqnarray} | (10) |
where
The weak gradient operator
\begin{eqnarray*} (\nabla_d v)|_T = \nabla_{d}(v|_T). \end{eqnarray*} |
We introduce the following bilinear form:
\begin{eqnarray*} a(v,w) = (\nabla_d v, \nabla_d w), \end{eqnarray*} |
the conforming DG algorithm to solve the problems (1) - (2) is given by
Conforming DG algorithm 1. Find
\begin{eqnarray} a(u_h, v_h) = (f, v_h), \; \; \forall \; v_h\in V^0_h, \end{eqnarray} | (11) |
where
We will prove the existence and uniqueness of the solution of equation (11). Firstly, we present the following two useful inequalities to derive the forthcoming analysis.
Lemma 3.1 (trace inequality). Let
\begin{eqnarray} \|\varphi\|_e^2\leq C(h_T^{-1}\|\varphi\|_T^2+h_T\|\nabla\varphi\|_T^2), \end{eqnarray} | (12) |
where
Lemma 3.2 (inverse inequality). Let
\begin{eqnarray} \|\nabla\varphi\|_T\leq C(n)h_T^{-1}\|\varphi\|_T,\; \; \forall T\in \mathcal{T}_h. \end{eqnarray} | (13) |
Then, we define the following semi-norms in the discontinuous finite element space
\begin{eqnarray} |\!|\!| v |\!|\!|^2& = &a(v, v) = \sum\limits_{T\in \mathcal{T}_h}\| \nabla_d v\|_T^2, \end{eqnarray} | (14) |
\begin{eqnarray} \| v \|_{1,h}^2& = &\sum\limits_{T\in \mathcal{T}_h}\|\nabla v\|_T^2+\sum\limits_{e\in\mathcal{E}_h^0}h_e^{-1}\|[\![ v]\! ]\|_e^2. \end{eqnarray} | (15) |
We have the equivalence between the semi-norms
Lemma 3.3. For any
\begin{eqnarray} C_1\|v\|_{1,h}\leq|\!|\!| v |\!|\!|\leq C_2\|v\|_{1,h}, \end{eqnarray} | (16) |
where
Proof. It follows from the definition of
\begin{eqnarray} \|\nabla_d v\|_{T_1}^2 & = & (\nabla_d v, \nabla_d v)_{T_1} = -(v, \nabla\cdot\nabla_d v)_{T_1} + \langle\{v\}\boldsymbol{n}, \nabla_d v\rangle_{\partial {T_1}}\\ & = & (\nabla v, \nabla_d v)_{T_1} - \langle (v-\{v\})\boldsymbol{n}, \nabla_d v\rangle_{\partial {T_1}}\\ &\leq& \|\nabla v\|_{T_1}\|\nabla_d v\|_{T_1}+\|(v-\{v\})\boldsymbol{n}\|_{\partial {T_1}}\|\nabla_d v\|_{\partial {T_1}}\\ &\leq&\|\nabla_d v\|_{T_1}(\|\nabla v\|_{T_1} + h_{T_1}^{-\frac{1}{2}}\|(v-\{v\})\boldsymbol{n}\|_{\partial {T_1}}). \end{eqnarray} | (17) |
For any
\begin{eqnarray*} (v-\{v\})|_e\boldsymbol{n}_1 & = & v|_{\partial T_1}\boldsymbol{n}_1-\frac{1}{2}(v|_{\partial T_1}+v|_{\partial T_2})\boldsymbol{n}_1\\ & = & \frac{1}{2}(v|_{\partial T_1}\boldsymbol{n}_1+v|_{\partial T_2}\boldsymbol{n}_2)\\ & = & \frac{1}{2}[\![ v]\! ]_e. \end{eqnarray*} |
Then we can get
\begin{eqnarray} \|(v-\{v\})\boldsymbol{n}\|_{\partial T_1}^2\leq\frac{1}{2}\sum\limits_{e\in\partial T_1}\|[\![ v]\! ]\|_e^2. \end{eqnarray} | (18) |
Substituting (18) into (17) gives
\begin{eqnarray*} \|\nabla_d v\|_{T_1}^2 &\leq& C_2\|\nabla_d v\|_{T_1}(\|\nabla v\|_{T_1}+\sum\limits_{e\in\partial T_1}h_e^{-\frac{1}{2}}\|[\![ v]\! ]\|_e), \end{eqnarray*} |
this completes the proof of the right-hand of (16).
To prove the left-hand of (16), we consider the subspace of
\begin{eqnarray*} D(k,T): = \{\bf{q}\in RT_k(T):\; \boldsymbol{q}\cdot\boldsymbol{n} = 0\; \text{on}\; \partial T\}. \end{eqnarray*} |
Note that
\begin{eqnarray} \|\nabla v\|_T = \sup\limits_{\boldsymbol{q}\in D(k,T)}\frac{(\nabla v,\boldsymbol{q})_T}{\|\boldsymbol{q}\|_T}. \end{eqnarray} | (19) |
Using the integration by parts, Cauchy-Schwarz inequality, the definition of
\begin{eqnarray*} (\nabla v,\boldsymbol{q})_T & = & -(v,\nabla\cdot\boldsymbol{q})_T + \langle v, \boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T}\\ & = & (\nabla_d v, \boldsymbol{q})_T-\langle \{v\},\boldsymbol{q}\cdot \boldsymbol{n}\rangle_{\partial T}\\ & = & (\nabla_d v, \boldsymbol{q})_T\\ &\leq& \|\nabla_d v\|_T\cdot\|\boldsymbol{q}\|_T, \end{eqnarray*} |
where we have used the fact that
\begin{eqnarray} \|\nabla v\|_T \leq \|\nabla_d v\|_T. \end{eqnarray} | (20) |
We define the space
\begin{eqnarray} \|[\![ v]\! ]\|_e = \sup\limits_{\boldsymbol{q}\in D_e(k,T)}\frac{\langle [\![ v]\! ], \boldsymbol{q}\cdot\boldsymbol{n}\rangle_e}{\|\boldsymbol{q}\cdot\boldsymbol{n}\|_e}. \end{eqnarray} | (21) |
Following the integration by parts and the definition of
\begin{eqnarray*} (\nabla_d v,\boldsymbol{q})_T = ( \nabla v, \boldsymbol{q} )_T - \langle v,\boldsymbol{q}\cdot\boldsymbol{n}\rangle_e + \langle \{v\}, \boldsymbol{q}\cdot\boldsymbol{n}\rangle_e. \end{eqnarray*} |
Together with (20), we obtain
\begin{eqnarray*} |\langle [\![ v]\! ], \boldsymbol{q}\cdot\boldsymbol{n}\rangle_e| & = &2|(\nabla_d v,\boldsymbol{q})_T - (\nabla v, \boldsymbol{q})_T|\\ &\leq& 2|(\nabla_d v,\boldsymbol{q})_T|+2|(\nabla v, \boldsymbol{q})_T|\\ &\leq& C(\|\nabla_d v\|_T\|\boldsymbol{q}\|_T+\|\nabla v\|_T\|\boldsymbol{q}\|_T)\\ &\leq& C\|\nabla_d v\|_T\|\boldsymbol{q}\|_T. \end{eqnarray*} |
Substituting the above inequality into (21), by the scaling argument [13], for such
\begin{eqnarray} \|[\![ v]\! ]\|_e \leq C\frac{\|\nabla_d v\|_T\|\boldsymbol{q}\|_T}{\|\boldsymbol{q}\cdot\boldsymbol{n}\|_e} \leq Ch^{\frac{1}{2}}\|\nabla_d v\|_T. \end{eqnarray} | (22) |
Combining (20) and (22) gives a proof of the left-hand of (16).
Lemma 3.4. The semi-norm
Proof. We shall only verify the positivity property for
The above two lemmas imply the well posedness of the scheme (11). We prove the existence and uniqueness of solution of the conforming DG method in Theorem 3.1.
Theorem 3.1. The conforming DG scheme (11) has and only has one solution.
Proof. To prove the scheme (11) is uniquely solvable, it suffices to verify that the homogeneous equation has zero as its unique solution. To this end, let
\begin{eqnarray*} a(u_h, u_h) = 0, \end{eqnarray*} |
which leads to
In this section, we will derive an error equation which will be used for the error estimates. For any
\begin{eqnarray} (\nabla\cdot\boldsymbol{q},v)_T = (\nabla\cdot\Pi_h\boldsymbol{q}, v)_T, \; \; \forall v \in Q_k(T). \end{eqnarray} | (23) |
For any
\begin{eqnarray} \|\Pi_h(\nabla w)-\nabla w\|\leq Ch^k\|w\|_{1+k}. \end{eqnarray} | (24) |
Moreover, it is easy to verify the following property holds true.
Lemma 4.1. For any
\begin{eqnarray} \sum\limits_{T\in\mathcal{T}_h}(-\nabla\cdot\boldsymbol{q}, v)_T = \sum\limits_{T\in\mathcal{T}_h}(\Pi_h\boldsymbol{q}, \nabla_d v)_T, \; \; \forall v\in V_h^0. \end{eqnarray} | (25) |
Proof.
\begin{eqnarray} \sum\limits_{T\in\mathcal{T}_h}\langle\{v\}, \Pi_h\boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T} = 0. \end{eqnarray} | (26) |
By the definition of
\begin{eqnarray*} \sum\limits_{T\in\mathcal{T}_h}(-\nabla\cdot\boldsymbol{q}, v)_T & = & \sum\limits_{T\in\mathcal{T}_h}(-\nabla\cdot\Pi_h\boldsymbol{q}, v)_T\\ & = & \sum\limits_{T\in\mathcal{T}_h}(-\nabla\cdot\Pi_h\boldsymbol{q}, v)_T+ \sum\limits_{T\in\mathcal{T}_h}\langle\{v\}, \Pi_h\boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T}\\ & = & \sum\limits_{T\in\mathcal{T}_h}(\Pi_h\boldsymbol{q}, \nabla_d v)_T. \end{eqnarray*} |
This completes the proof of the lemma.
Before establishing the error equation, we define a continuous finite element subspace of
\begin{eqnarray} \tilde{V}_{h} = \{v \in H^{1}(\Omega) :v|_T \in Q_k(T), \; \forall T \in \mathcal{T}_{h}\}. \end{eqnarray} | (27) |
so as a subspace of
\begin{eqnarray} \tilde{V}_h^0: = \{v\in\tilde{V}_h:v|_{\partial\Omega} = 0\}. \end{eqnarray} | (28) |
Lemma 4.2. For any
\begin{eqnarray*} \nabla_d v = \nabla v. \end{eqnarray*} |
Proof. By the definition of
\begin{eqnarray*} (\nabla_d v, \boldsymbol{q})_T & = & -(v, \nabla\cdot\boldsymbol{q})_T + \langle\{v\}, \boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T}\\ & = & -(v,\nabla\cdot\boldsymbol{q})_T + \langle v, \boldsymbol{q}\cdot\boldsymbol{n}\rangle_{\partial T}\\ & = & (\nabla v, \boldsymbol{q})_T, \end{eqnarray*} |
which gives
\begin{eqnarray*} (\nabla_d v-\nabla v, \boldsymbol{q})_T = 0, \; \; \forall \boldsymbol{q}\in RT_k(T). \end{eqnarray*} |
Letting
Let
\begin{eqnarray} \|I_hu-u\|\leq Ch^{k+1}\|u\|_{k+1}, \end{eqnarray} | (29) |
\begin{eqnarray} \|\nabla I_hu-\nabla u\|\leq Ch^k\|u\|_{k+1}. \end{eqnarray} | (30) |
It is obvious that
Lemma 4.3. Denote
\begin{eqnarray} a(e_h, v_h) = l_u( v_h), \end{eqnarray} | (31) |
where
\begin{eqnarray} l_u(v_h) = \sum\limits_{T\in{\mathcal{T}_h}}(\nabla I_{h} u-\Pi_h\nabla u, \nabla_d v_h). \end{eqnarray} | (32) |
Proof. Since
\begin{eqnarray*} \sum\limits_{T\in{\mathcal{T}_h}}(\nabla_d I_h u, \nabla_d v_h)_T & = & \sum\limits_{T\in{\mathcal{T}_h}}(\nabla I_h u, \nabla_d v_h)_T\\ & = & \sum\limits_{T\in{\mathcal{T}_h}}(\nabla I_h u -\Pi_h\nabla u + \Pi_h\nabla u, \nabla_d v_h)_T\\ & = & \sum\limits_{T\in{\mathcal{T}_h}}(\nabla I_h u -\Pi_h\nabla u, \nabla_d v_h)_T +\sum\limits_{T\in{\mathcal{T}_h}}(\Pi_h\nabla u, \nabla_d v_h)_T\\ & = & l_u(v_h) - \sum\limits_{T\in{\mathcal{T}_h}}(\nabla\cdot\nabla u, v_h)_T\\ & = & l_u(v_h) + (f,v_h). \end{eqnarray*} |
By the definition of the scheme (11), we have
\begin{eqnarray*} \sum\limits_{T\in{\mathcal{T}_h}}(\nabla_d I_h u - \nabla_d u_h, \nabla_d v_h)_T = l_u(v_h). \end{eqnarray*} |
This completes the proof of the lemma.
The goal of this section is to derive the error estimates in
Theorem 5.1. Let
\begin{eqnarray} |\!|\!|e_h|\!|\!| \leq Ch^k|u|_{k+1}. \end{eqnarray} | (33) |
Proof. Letting
\begin{eqnarray} |\!|\!|e_h|\!|\!|^2 = l_u(e_h). \end{eqnarray} | (34) |
From the Cauchy-Schwarz inequality, the triangle inequality, the definition of
\begin{eqnarray*} l_u(v_h)& = & \sum\limits_{T\in\mathcal{T}_h}(\nabla I_hu-\Pi_h(\nabla u), \nabla_d v_h)_T\\ &\leq& \sum\limits_{T\in\mathcal{T}_h}\|\nabla I_hu-\Pi_h(\nabla u)\|_T\|\nabla_d v_h\|_T\\ &\leq& \left(\sum\limits_{T\in\mathcal{T}_h}\|\nabla I_hu-\Pi_h(\nabla u)\|_T^2\right)^{\frac{1}{2}}\left(\sum\limits_{T\in\mathcal{T}_h}\|\nabla_d v_h\|_T^2\right)^{\frac{1}{2}}\\ & = & \left(\sum\limits_{T\in\mathcal{T}_h}\|\nabla I_hu-\nabla u+\nabla u-\Pi_h(\nabla u)\|_T^2\right)^{\frac{1}{2}}|\!|\!|v_h|\!|\!|\\ &\leq& \left(\sum\limits_{T\in\mathcal{T}_h}\|\nabla I_hu-\nabla u\|_T^2+\|\nabla u-\Pi_h(\nabla u)\|_T^2\right)^{\frac{1}{2}}|\!|\!|v_h|\!|\!|\\ &\leq& Ch^k|u|_{k+1}|\!|\!|v_h|\!|\!|. \end{eqnarray*} |
Then, we have
\begin{eqnarray} l_u(e_h) \leq Ch^k|u|_{k+1}|\!|\!|e_h|\!|\!|. \end{eqnarray} | (35) |
Substituting (35) to (34), we obtain
\begin{eqnarray*} |\!|\!|e_h|\!|\!|^2 \leq Ch^k|u|_{k+1}|\!|\!|e_h|\!|\!|, \end{eqnarray*} |
which completes the proof of the lemma.
It is obvious that
\begin{eqnarray} (\nabla\tilde{u}_h, \nabla v) = (f, v), \; \; \forall v\in\tilde{V}_h^0. \end{eqnarray} | (36) |
For any
\begin{eqnarray} (\nabla_d u_h - \nabla\tilde{u}_h, \nabla v) = 0, \; \; \forall v\in\tilde{V}_h^0. \end{eqnarray} | (37) |
In the rest of this section, we derive an optimal order error estimate for the conforming DG approximation (11) in
\begin{eqnarray} -\nabla\cdot(\nabla\Phi) = u_h-\tilde{u}_h, \; \; in\; \Omega. \end{eqnarray} | (38) |
Assume that the dual problem satisfies
\begin{eqnarray} \|\Phi\|_2 \leq C\|u_h-\tilde{u}_h\|. \end{eqnarray} | (39) |
In the following of this paper, we note
Theorem 5.2. Assume
\begin{eqnarray} \|u-u_h\|\leq C h^{k+1}|u|_{k+1}. \end{eqnarray} | (40) |
Proof. First, we shall derive the optimal order for
\begin{eqnarray} a(\Phi_h, v) = (\varepsilon_h, v), \; \; \forall v\in V_h^0. \end{eqnarray} | (41) |
Since
\begin{eqnarray*} (\nabla_d u_h-\nabla\tilde{u}_h, \nabla I_h\Phi) & = & 0,\\ \nabla_d I_h\Phi & = & \nabla I_h\Phi, \end{eqnarray*} |
which gives
\begin{eqnarray} (\nabla_d u_h-\nabla\tilde{u}_h, \nabla_d I_h\Phi) = 0. \end{eqnarray} | (42) |
Setting
\begin{eqnarray*} \|\varepsilon_h\|^2& = &a(\Phi_h, \varepsilon_h) = \sum\limits_{T\in\mathcal{T}_h}(\nabla_d \Phi_h, \nabla_d\varepsilon_h)_T\\ & = & \sum\limits_{T\in\mathcal{T}_h}(\nabla_d (\Phi_h-I_h\Phi), \nabla_d u_h -\nabla\tilde{u}_h)_T\\ &\leq&|\!|\!| \Phi_h-I_h\Phi |\!|\!|(|\!|\!|u_h-I_h u|\!|\!|+\|\nabla(I_h u-\tilde{u}_h)\|). \end{eqnarray*} |
Then, by the Cauchy-Schwarz inequality, (33) and (39), we obtain
\begin{eqnarray*} \|\varepsilon_h\|^2&\leq& Ch|\Phi|_2h^k|u|_{k+1} \leq Ch^{k+1}|u|_{k+1}\|\varepsilon_h\|, \end{eqnarray*} |
which gives
\begin{eqnarray} \|\varepsilon_h\| \leq Ch^{k+1}|u|_{k+1}. \end{eqnarray} | (43) |
Combining the error estimate of finite element solution, the triangle inequality and (43) yields (40), which completes the proof of the theorem.
In this section, we shall present some numerical results for the conforming discontinuous Galerkin method analyzed in the previous sections.
We solve the following Poisson equation on the unit square domain
\begin{align} -\Delta u & = 2 \pi^2 \sin(\pi x)\sin(\pi y) &&\hbox{in }\Omega && \end{align} | (44) |
\begin{align} u& = 0 &&\hbox{on }\partial\Omega. \end{align} | (45) |
The exact solution of the above problem is
We first use the
level | rate | rate | |||
by |
|||||
6 | 0.1996E-02 | 1.97 | 0.8887E-02 | 1.98 | 1024 |
7 | 0.5013E-03 | 1.99 | 0.2228E-02 | 2.00 | 4096 |
8 | 0.1255E-03 | 2.00 | 0.5574E-03 | 2.00 | 16384 |
by |
|||||
6 | 0.2427E-02 | 1.97 | 0.1027E+00 | 1.02 | 3072 |
7 | 0.6100E-03 | 1.99 | 0.5105E-01 | 1.01 | 12288 |
8 | 0.1527E-03 | 2.00 | 0.2546E-01 | 1.00 | 49152 |
by |
|||||
5 | 0.1533E-03 | 3.00 | 0.2042E-01 | 2.03 | 1536 |
6 | 0.1915E-04 | 3.00 | 0.5061E-02 | 2.01 | 6144 |
7 | 0.2394E-05 | 3.00 | 0.1260E-02 | 2.01 | 24576 |
by |
|||||
5 | 0.7959E-05 | 4.00 | 0.1965E-02 | 3.00 | 2560 |
6 | 0.4971E-06 | 4.00 | 0.2451E-03 | 3.00 | 10240 |
7 | 0.3140E-07 | 3.98 | 0.3059E-04 | 3.00 | 40960 |
by |
|||||
4 | 0.1055E-04 | 4.97 | 0.1421E-02 | 4.05 | 960 |
5 | 0.3314E-06 | 4.99 | 0.8735E-04 | 4.02 | 3840 |
6 | 0.1057E-07 | 4.97 | 0.5417E-05 | 4.01 | 15360 |
by |
|||||
2 | 0.2835E-02 | 6.24 | 0.1450E+00 | 5.49 | 84 |
3 | 0.4532E-04 | 5.97 | 0.4718E-02 | 4.94 | 336 |
4 | 0.7115E-06 | 5.99 | 0.1478E-03 | 5.00 | 1344 |
The same test case is also computed using the
level | rate | rate | |||
by |
|||||
6 | 0.4006E-03 | 1.99 | 0.2389E-02 | 1.99 | 4096 |
7 | 0.1003E-03 | 2.00 | 0.5982E-03 | 2.00 | 16384 |
8 | 0.2510E-04 | 2.00 | 0.1496E-03 | 2.00 | 65536 |
by |
|||||
6 | 0.2360E-04 | 2.99 | 0.3186E-02 | 1.99 | 9216 |
7 | 0.2953E-05 | 3.00 | 0.7976E-03 | 2.00 | 36864 |
8 | 0.3692E-06 | 3.00 | 0.1995E-03 | 2.00 | 147456 |
by |
|||||
5 | 0.1413E-04 | 4.08 | 0.1650E-02 | 2.97 | 4096 |
6 | 0.8676E-06 | 4.03 | 0.2072E-03 | 2.99 | 16384 |
7 | 0.5398E-07 | 4.01 | 0.2593E-04 | 3.00 | 65536 |
by |
|||||
3 | 0.2226E-02 | 4.59 | 0.5414E-01 | 3.52 | 400 |
4 | 0.9610E-04 | 4.53 | 0.3723E-02 | 3.86 | 1600 |
5 | 0.3279E-05 | 4.87 | 0.2392E-03 | 3.96 | 6400 |
To test the superconvergence of
\begin{align*} -\Delta u +u & = f &&\hbox{in }\Omega &&\\ u& = 0 &&\hbox{on }\partial\Omega, \end{align*} |
where
\begin{align} u = (x-x^2)(y-y^3). \end{align} | (46) |
Uniform square grids as shown in Figure 1 are used for numerical computation. The numerical results are listed in Table 3. Surprising, for this problem, the
level | rate | rate | |||
by |
|||||
3 | 0.8265E-02 | 1.06 | 0.4577E-01 | 1.14 | 16 |
4 | 0.2772E-02 | 1.58 | 0.1732E-01 | 1.40 | 64 |
5 | 0.7965E-03 | 1.80 | 0.6331E-02 | 1.45 | 256 |
6 | 0.2142E-03 | 1.90 | 0.2290E-02 | 1.47 | 1024 |
7 | 0.5564E-04 | 1.94 | 0.8213E-03 | 1.48 | 4096 |
8 | 0.1419E-04 | 1.97 | 0.2928E-03 | 1.49 | 16384 |
To test further the superconvergence of
\begin{align*} -\nabla(a \nabla u ) & = f &&\hbox{in }\Omega &&\\ u& = 0 &&\hbox{on }\partial\Omega, \end{align*} |
where
\begin{align} u = (x-x^3)(y^2-y^3). \end{align} | (47) |
Uniform square grids as shown in Figure 1 are used for computation. The numerical results are listed in Table 4. Surprising, again, the
level | rate | rate | |||
by |
|||||
3 | 0.4929E-02 | 0.97 | 0.5371E-01 | 0.80 | 16 |
4 | 0.1917E-02 | 1.36 | 0.2401E-01 | 1.16 | 64 |
5 | 0.6004E-03 | 1.67 | 0.9407E-02 | 1.35 | 256 |
6 | 0.1682E-03 | 1.84 | 0.3507E-02 | 1.42 | 1024 |
7 | 0.4457E-04 | 1.92 | 0.1275E-02 | 1.46 | 4096 |
8 | 0.1148E-04 | 1.96 | 0.4576E-03 | 1.48 | 16384 |
In this paper, we establish a new numerical approximation scheme based on the rectangular partition to solve second order elliptic equation. We derived the numerical scheme and then proved the optimal order of convergence of the error estimates in
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level | rate | rate | |||
by |
|||||
6 | 0.1996E-02 | 1.97 | 0.8887E-02 | 1.98 | 1024 |
7 | 0.5013E-03 | 1.99 | 0.2228E-02 | 2.00 | 4096 |
8 | 0.1255E-03 | 2.00 | 0.5574E-03 | 2.00 | 16384 |
by |
|||||
6 | 0.2427E-02 | 1.97 | 0.1027E+00 | 1.02 | 3072 |
7 | 0.6100E-03 | 1.99 | 0.5105E-01 | 1.01 | 12288 |
8 | 0.1527E-03 | 2.00 | 0.2546E-01 | 1.00 | 49152 |
by |
|||||
5 | 0.1533E-03 | 3.00 | 0.2042E-01 | 2.03 | 1536 |
6 | 0.1915E-04 | 3.00 | 0.5061E-02 | 2.01 | 6144 |
7 | 0.2394E-05 | 3.00 | 0.1260E-02 | 2.01 | 24576 |
by |
|||||
5 | 0.7959E-05 | 4.00 | 0.1965E-02 | 3.00 | 2560 |
6 | 0.4971E-06 | 4.00 | 0.2451E-03 | 3.00 | 10240 |
7 | 0.3140E-07 | 3.98 | 0.3059E-04 | 3.00 | 40960 |
by |
|||||
4 | 0.1055E-04 | 4.97 | 0.1421E-02 | 4.05 | 960 |
5 | 0.3314E-06 | 4.99 | 0.8735E-04 | 4.02 | 3840 |
6 | 0.1057E-07 | 4.97 | 0.5417E-05 | 4.01 | 15360 |
by |
|||||
2 | 0.2835E-02 | 6.24 | 0.1450E+00 | 5.49 | 84 |
3 | 0.4532E-04 | 5.97 | 0.4718E-02 | 4.94 | 336 |
4 | 0.7115E-06 | 5.99 | 0.1478E-03 | 5.00 | 1344 |
level | rate | rate | |||
by |
|||||
6 | 0.4006E-03 | 1.99 | 0.2389E-02 | 1.99 | 4096 |
7 | 0.1003E-03 | 2.00 | 0.5982E-03 | 2.00 | 16384 |
8 | 0.2510E-04 | 2.00 | 0.1496E-03 | 2.00 | 65536 |
by |
|||||
6 | 0.2360E-04 | 2.99 | 0.3186E-02 | 1.99 | 9216 |
7 | 0.2953E-05 | 3.00 | 0.7976E-03 | 2.00 | 36864 |
8 | 0.3692E-06 | 3.00 | 0.1995E-03 | 2.00 | 147456 |
by |
|||||
5 | 0.1413E-04 | 4.08 | 0.1650E-02 | 2.97 | 4096 |
6 | 0.8676E-06 | 4.03 | 0.2072E-03 | 2.99 | 16384 |
7 | 0.5398E-07 | 4.01 | 0.2593E-04 | 3.00 | 65536 |
by |
|||||
3 | 0.2226E-02 | 4.59 | 0.5414E-01 | 3.52 | 400 |
4 | 0.9610E-04 | 4.53 | 0.3723E-02 | 3.86 | 1600 |
5 | 0.3279E-05 | 4.87 | 0.2392E-03 | 3.96 | 6400 |
level | rate | rate | |||
by |
|||||
3 | 0.8265E-02 | 1.06 | 0.4577E-01 | 1.14 | 16 |
4 | 0.2772E-02 | 1.58 | 0.1732E-01 | 1.40 | 64 |
5 | 0.7965E-03 | 1.80 | 0.6331E-02 | 1.45 | 256 |
6 | 0.2142E-03 | 1.90 | 0.2290E-02 | 1.47 | 1024 |
7 | 0.5564E-04 | 1.94 | 0.8213E-03 | 1.48 | 4096 |
8 | 0.1419E-04 | 1.97 | 0.2928E-03 | 1.49 | 16384 |
level | rate | rate | |||
by |
|||||
3 | 0.4929E-02 | 0.97 | 0.5371E-01 | 0.80 | 16 |
4 | 0.1917E-02 | 1.36 | 0.2401E-01 | 1.16 | 64 |
5 | 0.6004E-03 | 1.67 | 0.9407E-02 | 1.35 | 256 |
6 | 0.1682E-03 | 1.84 | 0.3507E-02 | 1.42 | 1024 |
7 | 0.4457E-04 | 1.92 | 0.1275E-02 | 1.46 | 4096 |
8 | 0.1148E-04 | 1.96 | 0.4576E-03 | 1.48 | 16384 |
level | rate | rate | |||
by |
|||||
6 | 0.1996E-02 | 1.97 | 0.8887E-02 | 1.98 | 1024 |
7 | 0.5013E-03 | 1.99 | 0.2228E-02 | 2.00 | 4096 |
8 | 0.1255E-03 | 2.00 | 0.5574E-03 | 2.00 | 16384 |
by |
|||||
6 | 0.2427E-02 | 1.97 | 0.1027E+00 | 1.02 | 3072 |
7 | 0.6100E-03 | 1.99 | 0.5105E-01 | 1.01 | 12288 |
8 | 0.1527E-03 | 2.00 | 0.2546E-01 | 1.00 | 49152 |
by |
|||||
5 | 0.1533E-03 | 3.00 | 0.2042E-01 | 2.03 | 1536 |
6 | 0.1915E-04 | 3.00 | 0.5061E-02 | 2.01 | 6144 |
7 | 0.2394E-05 | 3.00 | 0.1260E-02 | 2.01 | 24576 |
by |
|||||
5 | 0.7959E-05 | 4.00 | 0.1965E-02 | 3.00 | 2560 |
6 | 0.4971E-06 | 4.00 | 0.2451E-03 | 3.00 | 10240 |
7 | 0.3140E-07 | 3.98 | 0.3059E-04 | 3.00 | 40960 |
by |
|||||
4 | 0.1055E-04 | 4.97 | 0.1421E-02 | 4.05 | 960 |
5 | 0.3314E-06 | 4.99 | 0.8735E-04 | 4.02 | 3840 |
6 | 0.1057E-07 | 4.97 | 0.5417E-05 | 4.01 | 15360 |
by |
|||||
2 | 0.2835E-02 | 6.24 | 0.1450E+00 | 5.49 | 84 |
3 | 0.4532E-04 | 5.97 | 0.4718E-02 | 4.94 | 336 |
4 | 0.7115E-06 | 5.99 | 0.1478E-03 | 5.00 | 1344 |
level | rate | rate | |||
by |
|||||
6 | 0.4006E-03 | 1.99 | 0.2389E-02 | 1.99 | 4096 |
7 | 0.1003E-03 | 2.00 | 0.5982E-03 | 2.00 | 16384 |
8 | 0.2510E-04 | 2.00 | 0.1496E-03 | 2.00 | 65536 |
by |
|||||
6 | 0.2360E-04 | 2.99 | 0.3186E-02 | 1.99 | 9216 |
7 | 0.2953E-05 | 3.00 | 0.7976E-03 | 2.00 | 36864 |
8 | 0.3692E-06 | 3.00 | 0.1995E-03 | 2.00 | 147456 |
by |
|||||
5 | 0.1413E-04 | 4.08 | 0.1650E-02 | 2.97 | 4096 |
6 | 0.8676E-06 | 4.03 | 0.2072E-03 | 2.99 | 16384 |
7 | 0.5398E-07 | 4.01 | 0.2593E-04 | 3.00 | 65536 |
by |
|||||
3 | 0.2226E-02 | 4.59 | 0.5414E-01 | 3.52 | 400 |
4 | 0.9610E-04 | 4.53 | 0.3723E-02 | 3.86 | 1600 |
5 | 0.3279E-05 | 4.87 | 0.2392E-03 | 3.96 | 6400 |
level | rate | rate | |||
by |
|||||
3 | 0.8265E-02 | 1.06 | 0.4577E-01 | 1.14 | 16 |
4 | 0.2772E-02 | 1.58 | 0.1732E-01 | 1.40 | 64 |
5 | 0.7965E-03 | 1.80 | 0.6331E-02 | 1.45 | 256 |
6 | 0.2142E-03 | 1.90 | 0.2290E-02 | 1.47 | 1024 |
7 | 0.5564E-04 | 1.94 | 0.8213E-03 | 1.48 | 4096 |
8 | 0.1419E-04 | 1.97 | 0.2928E-03 | 1.49 | 16384 |
level | rate | rate | |||
by |
|||||
3 | 0.4929E-02 | 0.97 | 0.5371E-01 | 0.80 | 16 |
4 | 0.1917E-02 | 1.36 | 0.2401E-01 | 1.16 | 64 |
5 | 0.6004E-03 | 1.67 | 0.9407E-02 | 1.35 | 256 |
6 | 0.1682E-03 | 1.84 | 0.3507E-02 | 1.42 | 1024 |
7 | 0.4457E-04 | 1.92 | 0.1275E-02 | 1.46 | 4096 |
8 | 0.1148E-04 | 1.96 | 0.4576E-03 | 1.48 | 16384 |