A new approach of superconvergence analysis of nonconforming Wilson finite element for semi-linear parabolic problem
Introduction
We consider the following semi-linear parabolic problem [1]: where is a rectangle with boundary ∂Ω, , , , , and are smooth functions. Assume that there exists a positive constant M such that
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 -norm is just of order 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 . 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 . Recently, by adding an interior penalty term to the traditional energy norm, a DGM was discussed in [12] when (i.e., the linear case), an 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 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 , order and order respectively in the modified energy norm (which is bigger than that of the traditional broken -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 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 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 are provided to confirm our theoretical analysis and the efficiency of the proposed methods.
Throughout this paper, denotes the standard Sobolev space with norm and semi-norm , and denotes the standard -inner product. When , we simply write and as and , respectively. We define the space with the norm (for short ) when , and if , the integral is replaced by the essential supremum. (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.
Section snippets
Wilson element and some estimates
Let us denote by a family of regular subdivision of Ω with mesh size h, the set of all edges of element of and the common edge with size shared by two elements K and . The jump and the average of a piecewise smooth function f over E are defined by and , respectively.
The finite element on each is defined as follows: where ,
Superconvergent analysis of semi-discrete scheme
The weak form of problem (1) is: to find , such that
We consider the semi-discrete approximation to (12) as: to find , such that
We introduce the DGM as: to find , such that Theorem 3.1 Let u and be the solutions of (1) and (14), respectively. Assume that , then for sufficiently small h,
Superconvergent analysis of linearized Backward Euler fully-discrete scheme
For some positive integer N, let be a given partition of the time interval with step length and . For a smooth function ϕ on , denote and .
We introduce the linearized Backward Euler fully-discrete scheme of (12) as: to find , such that for , , Theorem 4.1 Let and be the solutions of (1) and (23), respectively. Assume that ,
Superconvergent analysis of BDF2 fully-discrete scheme
In this section, we develop a BDF2 scheme to the system (12): find , such that for , ,
This scheme is not self starting, and the first step is given by the following predictor corrector method: to find , such that in which is derived by with the value . Obviously, for the above
Numerical experiments
In this section, we present the following two numerical examples to demonstrate the theoretical analysis. In the computation, we set the domain and the final time , respectively. We consider the problem where g and are computed from the exact solution
Example 1 .
Example 2 .
For simplicity, the domain Ω is divided into families of square meshes. The numerical results of semi-discrete scheme
Acknowledgement
This work is supported by the National Natural Science Foundation of China (No. 11671105).
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