Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations

1.   Introduction

In this paper, we consider the following semilinear parabolic integro-differential equations:

where $\Omega\subset{\mathbf{R}^2}$ is a rectangle with boundary $\partial \Omega$, $X = (x,y)$, $J = (0,T]$, $f(\cdot)$ is twice continuously differentiable and $u_{0}$ is a given function. We assume that

There exists a lot of numerical methods for solving nonlinear partial differential equations in the literature. For example, Cannon and Lin [1] derived a priori error estimates of semidiscrete and Crank-Nicolson finite element approximations to the solution of the nonlinear diffusion equations with memory. Eriksson and Johnson [3] used adaptive finite element method to solve nonlinear parabolic problems. Moore [9] considered a posteriori error estimates for semi- and fully discrete finite element methods using a degree polynomial basis for solving nonlinear parabolic equations. Garcia [4] discussed a priori error estimates of fully discrete Raviart-Thomas mixed finite element scheme for nonlinear parabolic equations.

Two-grid method was first proposed by Xu [11,12] as an efficient discretization technique for solving the nonlinear and the nonsymmetric problems. Liu et al. exploited the two kinds of two-grid algorithms for finite difference solutions of semilinear parabolic equations in [8]. Chen et al. [2] presented a two-grid scheme of mixed finite element method for fully nonlinear reaction-diffusion equations. Hou et al. [5] presented a two-grid method of $P_0^2$-$P_1$ mixed finite element method combined with Crank-Nicolson scheme for a class of nonlinear parabolic equations. Shi and Mu [10] discussed some superclose results of a two-grid finite element method for semilinear parabolic equations. Yang and Xing [13] discussed the convergence of two-grid discontinuous Galerkin scheme for a kind of nonlinear parabolic problems.

This paper is motivated by the ideas of the works [10,11], we present a two-grid scheme for semilinear parabolic integro-differential equations discretized by finite element method combined with Crank-Nicolson scheme. We mainly discuss the superclose estimates between the numerical solution and the interpolation.

The plan of this paper is as follows. In Section 2, we give the Crank-Nicolson scheme and deduce the superclose result of order $\mathcal{O}(h^2+(\triangle t)^2)$ in the $H^1$-norm. In Section 3, we present the two-grid method and derive the superclose estimates of order $\mathcal{O}(H^2+(\triangle t)^2)$ and order $\mathcal{O}(h^2+H^4+(\triangle t)^2)$, respectively. In Section 4, we present a numerical example to demonstrate the effectiveness of our method.

2.   Superclose analysis of the Crank-Nicolson scheme

We adopt the standard notation $W^{m,p}(\Omega)$ for Sobolev spaces on $\Omega$ with a norm $\|\cdot\|_{m,p}$ given by $\| v \|_{m,p}^{p} = \sum\limits_{|\alpha|\leq m}\| D^\alpha v\|_{L^{p}(\Omega)}^{p}$, a semi-norm $|\cdot|_{m,p}$ given by $| v|_{m,p}^{p} = \sum\limits_{|\alpha| = m}\| D^\alpha v\|_{L^{p}(\Omega)}^{p}$. We set $W_0^{m,p}(\Omega) = \{v\in W^{m,p}(\Omega): v|_{\partial \Omega} = 0\}$. For $p = 2$, we denote $H^m(\Omega) = W^{m,2}(\Omega),\ H_0^m(\Omega) = W_0^{m,2}(\Omega)$, and $\|\cdot\|_{m} = \|\cdot\|_{m,2},\ \|\cdot\| = \|\cdot\|_{0,2}$.

We denote by $L^s(J;W^{m,p}(\Omega))$ the Banach space of all $L^s$ integrable functions from $J$ into $W^{m,p}(\Omega)$ with norm $\|v\|_{L^s(J;W^{m,p}(\Omega))} = \Big(\int_0^T||v||_{W^{m,p}(\Omega)}^sdt\Big)^{\frac{1}{s}}\ \rm{for}\ s\in [1,\infty),$ and the standard modification for $s = \infty$. For simplicity of presentation, we denote $\| v\|_{L^s(J;W^{m,p}(\Omega))}$ by $\| v\|_{L^s(W^{m,p})}$. Similarly, one can define the spaces $H^1(J;W^{m,p}(\Omega))$. In addition $C$ denotes a general positive constant independent of $h$ and $\triangle t$, where $h$ is the spatial mesh-size, and $\triangle t$ is the time step.

Let $\mathcal{T}_{h}$ be a uniform rectangular partition of $\Omega$ with mesh size $h$. $V_{h}$ be the bilinear finite element space with vanishes on $\partial\Omega$. Let $I_{h}$ and $R_{h}$ be the associated interpolation and Ritz projection operators on $V_{h}$, respectively(see[10]).

Then, for $u\in H^1_{0}(\Omega)\cap H^3(\Omega)$, from [10] we know that

The weak formulation of (1) is to find $u:J\rightarrow H^1_{0}(\Omega)$, such that

Let $\{t_{n}\arrowvert t_{n} = n\triangle t; 0\leq n\leq N\}$ be a uniform partition in time with time step $\triangle t$, $u^{n} = u(X,t_{n})$ and $t_{n-1/2} = (t_{n-1}+t_n)/2$. For a sequence of functions $\{\phi^n\}^N_{n = 0}$, we denote $d_t\phi^n = (\phi^n-\phi^{n-1})/\triangle t$, then the Crank-Nicolson scheme of (1) is to find $u^n_{h}\in V_{h}$ for $n = 1,2,...,N$, such that

For the proof of existence and uniqueness of the solution for the nonlinear algebraic problem (9)-(10), please refer to [6].

Theorem 2.1. Let $u$ and $u^n_{h}$ be the solutions of (8) and (9), respectively. Assume that $u\in L^\infty(H^3)$, $u_t\in L^2(H^2)$, $u_{tt}\in L^2(L^2)$, $u_{ttt}\in L^2(L^2)$, $\nabla u_{tt}\in L^2(H^2)\cap L^{\infty}(L^2)$ and $\nabla u_{ttt}\in L^2(L^2)$, then, for $n = 1,2,\ldots,N$, we have

Proof. Letting $t = t_{n-1/2}$ and $v = v_h$ in (8), we get

where $R^n_1 = d_tu^n-u^{n-1/2}_t$, $R^n_2 = \frac{u^n+u^{n-1}}{2}-u^{n-1/2}$.

Setting $u^n-u^n_h = u^n-R_hu^n+R_hu^n-u^n_h: = \eta^n+\xi^n$. Subtracting (9) from (12), with the help of (7), we have

Selecting $v_h = d_t\xi^n$ in (13), noting that

then multiplying (13) by 2$\triangle t$ and summing from $n = 1,...,l(1\leq l \leq N)$, we conclude that

where we used (7) and $\xi^0 = 0$.

Now, we estimate the right-hand terms of (15). For $I_1$, it is easy to check that

which together with Cauchy inequality, Young's inequality and $(4)$ yields

For $I_2$, we decompose it as

Using Cauchy inequality and Young's inequality, we see that

and

Thus, we get

For $I_3$, by virtue of Cauchy inequality and Young's inequality, we have

where we used the following estimate

where $\phi^n$ is located between $t_{n}$ and $t_{n-1/2}$, $\psi^n$ is located between $t_{n-1/2}$ and $t_{n-1}$, $\lambda^n$ is located between $\phi^n$ and $\psi^n$, and

Next, we estimate $I_4$. By use of mean value theorem and the assumption on $f$, we conclude that

where we also used

and

where $\theta^n_1$ is located between $t_{n-1}$ and $t_{n-1/2}$, $\theta^n_2$ is located between $t_{n-1/2}$ and $t_{n}$, $\theta^n_3$ is located between $\theta^n_1$ and $\theta^n_2$, $\lambda^n_1$ is located between $u^{n-1}$ and $u^{n-1/2}$, $\lambda^n_2$ is located between $u^{n-1/2}$ and $u^{n}$, $\lambda^n_3$ is located between $\lambda^n_1$ and $\lambda^n_2$.

Using (4), mean value theorem and the assumption on $f$, we easily get

By use of Cauchy inequality, Young's inequality and (24)-(25), we derive

For $I_5$, from Cauchy inequality, Young's inequality and

we have

where $\rho^n$ is located between $t_{n-1}$ and $t_{n}$.

Notice that

Using Taylor expansion, we know that

where $t_{n}<\beta^n<t_{n+1/2}<\gamma^{n}< t_{n+1}$.

Using Cauchy inequality, Young's inequality, (29), (30) and mean value theorem, we get

where $\lambda^l$ is located between $t_{l-1}$ and $t_{l}$, $\delta^n$ is located between $t_{n-1/2}$ and $t_{n+1/2}$.

For $I_6$, combining Cauchy inequality, Young's inequality, (28) with (31)-(32), we derive

Now, substituting the estimates for $I_1$-$I_6$ into (15), then applying discrete Gronwall's lemma, for sufficiently small $\triangle t$, we have

which together with (5), Poincare's inequality and triangle inequality yields (11). We complete the proof of the theorem.

3.   Superclose analysis of the two-grid scheme

In this section, we present the main algorithm of the paper, which has the following two steps:

Step 1. On the coarse grid $\mathcal{T}_H$, compute $u^1_H\in V_H$ to satisfy the following original nonlinear system:

For $n = 1,...,N-1$, compute $u^{n+1}_H\in V_H$ to satisfy the following linear system:

Step 2. On the fine grid $\mathcal{T}_h$, for $n = 1,...,N$, compute $\widetilde{u}^n_h\in V_h$ to satisfy the following linear system:

Now, we shall discuss the superclose estimates of the above two-grid algorithm in the following theorem.

Theorem 3.1. Let $u$, $u^n_{H}$ and $\widetilde{u}^n_{h}$ be the solutions of (8), (35)-(37) and (35)-(39), respectively. Then under the conditions of Theorem 2.1, for $n = 1,2,\ldots,N$, we have

Proof. Setting $u^n-u^n_H = u^n-R_Hu^n+R_Hu^n-u^n_H: = \omega^n+\varphi^n$. From Theorem 2.1, (40) is obvious for $n = 1$.

For $n = 1,...,N-1$, letting $t = t_{n}$ and $v = v_H$ in (8), we get

where $R^n_3 = \frac{u^{n+1}-u^{n-1}}{2\triangle t}-u^{n}_t$, $R^n_4 = \frac{u^{n+1}+u^{n-1}}{2}-u^{n}$.

Subtracting (37) from (42), with the help of (7), we have

Selecting $v_H = \frac{\varphi^{n+1}-\varphi^{n-1}}{\triangle t}$ in (43), using $\varphi^0 = 0$ and the equality

then multiplying the resulting equation by 2$\triangle t$ and summing from $n = 1,...,l(1\leq l \leq N-1)$, we conclude that

where we used (7) and $\varphi^0 = 0$.

Similar to the estimates of $I_1$-$I_6$, we can estimate $D_1$-$D_6$ as

At last, for $D_7$, using (5), (40) and triangle inequality, we see that

Now, substituting the estimates for $D_1$-$D_7$ into (45), then applying discrete Gronwall's lemma, for sufficiently small $\triangle t$, we have

which together with (5), Poincare's inequality and triangle inequality yields

Using Taylor expansion, we have

where $\alpha^n$ is located between $u^n$ and $u^n_{H}$.

Setting $R_hu^n-\widetilde{u}^n_h: = \widetilde{\xi}^n$. Subtracting (38) from (12), similar to (15), we conclude that

Now, we estimate $B_1$-$B_7$, respectively. For $B_3$, similar to (24), we know that

For $B_4$, we find from Cauchy inequality and the assumption on $f$ that

Using Young's inequality and (4), we know that

Combining (40), Cauchy inequality, Young's inequality and interpolation theory with $H^1\hookrightarrow L^4$, we derive

Now, from (59)-(60), we find that

Similarly, we can estimate $B_5$ as

Similar to (16), (21), (27) and (33), we easily have

It follows from (56)-(57), (61)-(63) and Poincare's inequality that

Thus, for sufficiently small $\triangle t$, using discrete Gronwall's lemma and Poincare's inequality, we arrive at

which together with (5) and triangle inequality yields

The proof is complete.

4.   Numerical experiments

In this section, we are going to validate the superclose estimates for two-grid discretization method for semilinear parabolic integro-differential equations by a concrete numerical example.

We consider the following semi-linear parabolic integro-differential equations

where $\Omega = (0,1)^2$ and $J = (0,1]$. We choose $u(X,t) = \sin(\pi t)\sin(\pi x_1)\sin(\pi x_2)$ as the exact solution. Then, the explicit formulation of $g(X,t)$ is

We first test the example for the Crank-Nicolson scheme. The error and the convergence order of $\|u_h^n-I_h u^n\|_1$ at $t = 0.125$ with $h = \Delta t$ are presented in Table 1. Obviously, it is the same with the result in Theorem 2.1. Next, the two-grid scheme is tested. The error and the convergence order of $\|u_H^n-I_H u^n\|_1$ and $\|\widetilde{u}_h^n-I_h u^n\|_1$ are provided in Table 2 and Table 3. We find from these two tables that the result coincides with that in Theorem 3.1. Finally, we show the efficiency of the two-grid method by comparing the cpu time in Table 4.