A Priori and a Posteriori Error Estimates of a WOPSIP DG Method for the Heat Equation

Yuping Zeng,Huijian Zhu,Kunwen Wen,Fen Liang

doi:10.1155/2020/7525676

Abstract

We introduce and analyze a weakly overpenalized symmetric interior penalty method for solving the heat equation. We first provide optimal a priori error estimates in the energy norm for the fully discrete scheme with backward Euler time-stepping. In addition, we apply elliptic reconstruction techniques to derive a posteriori error estimators, which can be used to design adaptive algorithms. Finally, we present two numerical experiments to validate our theoretical analysis.

Highlights

Let Ω ⊂ R2 be a bounded polygonal domain with Lipschitz boundary zΩ, we consider the following heat equation: ztu − Δu f, in Ω ×(0, T], u 0, on zΩ ×(0, T], (1)u(·, 0) u0, where T > 0 is finite time, f ∈ L2(0, T; L2(Ω)) is the source term, and u0 ∈ L2(Ω) stands for the initial data. ere has been much research on the a priori and a posteriori error estimates of finite element methods (FEM) for parabolic equations
E weakly overpenalized symmetric interior penalty (WOPSIP) method is a kind of nonconsistent discontinuous Galerkin (DG) scheme, which was initially proposed in [25] for solving the second order elliptic equation, therein a priori error estimates were obtained
The space variable is approximated by the WOPSIP method, and time variable is discretized by the backward Euler scheme

Summary

Introduction

Let Ω ⊂ R2 be a bounded polygonal domain with Lipschitz boundary zΩ, we consider the following heat equation: ztu − Δu f, in Ω ×(0, T], u 0, on zΩ ×(0, T],. Ern et al [22, 23] have developed a posteriori error estimates for the parabolic problem with DG discretization in time (see [24] for a posteriori error estimates of nonconforming Crouzeix–Raviart FEM for the heat equation). E WOPSIP method is a kind of nonconsistent discontinuous Galerkin (DG) scheme, which was initially proposed in [25] for solving the second order elliptic equation, therein a priori error estimates were obtained. We shall give a Mathematical Problems in Engineering detailed a priori and a posteriori error estimates In this case, one may come across a difficulty that stems from the nonconsistency of the numerical method (for more details, see (34) in eorem 2). We provide some numerical results to validate theoretical analysis of a priori error estimates

Preliminaries and WOPSIP Method

Stability and A Priori Error Analysis

A Posteriori Error Analysis

Numerical Experiments

Conclusion and Future Work