Weak Galerkin and Weak Group Finite Element Methods for 2-D Burgers' Problem

Abstract

In this paper, we apply weak Galerkin finite element method (WGFEM) and weak group finite element method (WGrFEM) for 2-D Burgers' problem by using the weak functions and their corresponding discrete weak derivatives in standard weak form. We consider the fully formulation using the backward-difference form for the time variable and we prove the stability and an error estimate for these methods. The numerical examples of our methods have been carried out through implementation in MATLAB programs and compared with the exact solutions and other available literature to illustrate the efficiency of the proposed methods, the WGFEM and the WGrFEM provide convergent approximations and handles the equation well in different cases, the results obtained of our methods are very acceptable and competent more than the results available in the other literature. Moreover, a WGrFEM is displayed to be better than the GFEM.

Numerical Analysis of Least-Squares Group Finite Element Method for Coupled Burgers' Problem

“…The group finite element method (GrFEM), also known as product approximation is a finite element (FE) technique for types of nonlinear PDEs. Experiments with the GrFEM have shown an increase in economy and in the nodal accuracy compared to FE solutions of the Burgers' equations [7][8][9] . In this paper, the least squares group finite element method for 2-D coupled Burgers' problem with a fully-discrete approximation for the time variable is presented.…”