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26-07-2017 | Issue 3/2018

Journal of Scientific Computing 3/2018

A Systematic Study on Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

Journal:
Journal of Scientific Computing > Issue 3/2018
Authors:
Junping Wang, Ruishu Wang, Qilong Zhai, Ran Zhang
Important notes
J. Wang: The research of Wang was supported by the NSF IR/D program, while working at National Science Foundation. However, any opinion, finding, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.
R. Zhang: The research of this author was supported in part by China Natural National Science Foundation (U1530116, 91630201, 11471141, J1310022), and by the Program for Cheung Kong Scholars of Ministry of Education of China, Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education, Jilin University, Changchun, 130012, P.R. China.

Abstract

This article provides a systematic study for the weak Galerkin (WG) finite element method for second order elliptic problems by exploring polynomial approximations with various degrees for each local element. A typical local WG element is of the form \(P_k(T)\times P_j(\partial T)\Vert P_\ell (T)^2\), where \(k\ge 1\) is the degree of polynomials in the interior of the element T, \(j\ge 0\) is the degree of polynomials on the boundary of T, and \(\ell \ge 0\) is the degree of polynomials employed in the computation of weak gradients or weak first order partial derivatives. A general framework of stability and error estimate is developed for the corresponding numerical solutions. Numerical results are presented to confirm the theoretical results. The work reveals some previously undiscovered strengths of the WG method for second order elliptic problems, and the results are expected to be generalizable to other type of partial differential equations.

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