A new approach of superconvergence analysis of nonconforming Wilson finite element for semi-linear parabolic problem

Abstract

In this paper, the discontinuous Galerkin method (DGM) of nonconforming Wilson element is studied for the semi-linear parabolic problem. The global superconvergence with respect to the mesh size are derived in the modified $H^1$-norm for the semi-discrete scheme and two fully discrete schemes, in which the usual extrapolation and interpolation post-processing approaches are not involved, and the error estimates are one order higher than that of the traditional Galerkin finite element method (FEM). Therefore, the corresponding results in the existing literature are improved. Finally, some numerical results are provided to confirm the theoretical analysis.

Introduction

We consider the following semi-linear parabolic problem [1]:

$$\{\begin{array}{cc}{u}_t-\mathrm{\Delta }u=f(u),\hfill & (X,t)\in \mathrm{\Omega }\times J,\hfill \\ u(X,t)=0,\hfill & (X,t)\in \partial \mathrm{\Omega }\times J,\hfill \\ u(X,0)=u_0(X),\hfill & X\in \mathrm{\Omega },\hfill \end{array}$$

where $\mathrm{\Omega }\subset R^2$ is a rectangle with boundary ∂Ω, $J=(0,T]$, $0<T<+\infty$, $X=(x,y)$, $u_t=\frac{\partial u}{\partial t}$, $u_0(X)$ and $f(u)$ are smooth functions. Assume that there exists a positive constant M such that

$$|f^{\prime }(u)|\le M,\forall u\in R.$$

There have been a lot of studies about the theoretical analysis and numerical simulations of FEMs for the parabolic problem (1) in [1]. In particular, the superconvergence behavior of the different approximation schemes were investigated throughly, such as the Galerkin FEM [2], mixed FEMs [3], [4], two-grid FEMs [5], [6], weak Galerkin FEM [7] and so on.

For the nonconforming Wilson element, based on a counter example, Shi et al. [8] proved that optimal convergent result in the broken $H^1$-norm is just of order $O(h)$ and can not be improved anymore even the exact solution of the considered problem is smooth enough, although its interpolation error can reach order of $O(h^2)$. Later on, Chen et al. [9] and Shi et al. [10] deduced the superconvergent behavior of the gradient errors on each center of element of the subdivision.

In order to get the global superconvergence, Lin et al. [11] applied the integral identity technique to the conforming part of Wilson element, and used the interpolation post-processing or extrapolation approach to yield the desired results of order $O(h^{2+\gamma })(\gamma \ge 0)$. Recently, by adding an interior penalty term to the traditional energy norm, a DGM was discussed in [12] when $f(u)=f(x)$ (i.e., the linear case), an $O(h^2)$ order error estimate was obtained only for the semi-discrete scheme. However, the proof of the main result Theorem 3.1 in [12] is not correct and needs to be reconsidered. Thus, how to derive an $O(h^2)$ order error estimate of Wilson element for parabolic problem (1) still remains open.

Motivated by [12], [6] and [13], [14], [15], our main aim of this paper is to fill this gap. We will investigate the semi-discrete, linearized backward Euler and second order fully-discrete schemes for problem (1) with DGM of Wilson element, and derive the corresponding superconvergence of order $O(h^2)$, order $O(h^2+\tau )$ and order $O(h^2+{\tau }^2)$ respectively in the modified energy norm (which is bigger than that of the traditional broken $H^1$-norm) without efforts on the analysis of the above mentioned approaches in [9] and [11]. It is worthy to mention that the analysis of this paper provides a new approach to deal with the superconvergent error estimates of other PDEs with DGM of Wilson element.

The rest of this paper is organized as follows. In section 2, we introduce the Wilson element and some known estimates in the modified norm. In section 3, we present the DGM for problem (1), and prove the superconvergent error estimate of order $O(h^2)$ for semi-discrete scheme which corrects the proof of [12]. In section 4, we develop a linearized backward Euler scheme, in which only a linear system needs to be solved at each time step. Then, we prove the superconvergent behavior of order $O(h^2+\tau )$ for this scheme. In section 5, we will present a BDF2 scheme and discuss its superconvergence. In the last section, numerical examples for both two kinds of $f(u)$ are provided to confirm our theoretical analysis and the efficiency of the proposed methods.

Throughout this paper, $W^{m,p}(\mathrm{\Omega })$ denotes the standard Sobolev space with norm ${\Vert \cdot \Vert }_{m,p}$ and semi-norm $|\cdot {|}_{m,p}$, and $(\cdot ,\cdot )$ denotes the standard $L^2$-inner product. When $p=2$, we simply write $|\cdot {|}_{m,p}$ and ${\Vert \cdot \Vert }_{m,p}$ as $|\cdot {|}_m$ and ${\Vert \cdot \Vert }_m$, respectively. We define the space $L^P(J;Y)$ with the norm ${\Vert f\Vert }_{L^p(J;Y)}={({\int }_J{\Vert f(\cdot ,t)\Vert }_Y^{p}dt)}^{\frac{1}{p}}$ (for short ${\Vert f\Vert }_{L^p(Y)}$) when $J=(0,T]$, and if $p=\infty$, the integral is replaced by the essential supremum. $C>0$ (with or without subscript) denotes a generic constant independent of the mesh size h, time step size τ and time level n, but may take different values at different places, ϵ represents a small positive constant in the ϵ-Young inequality.