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

• Published in: Journal of scientific computing Vol. 74; no. 3; pp. 1369 - 1396
• Main Authors: Wang, Junping, Wang, Ruishu, Zhai, Qilong, Zhang, Ran
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.03.2018; Springer Nature B.V
• Subjects: Algorithms; Computational Mathematics and Numerical Analysis; Estimates; Finite element analysis; Finite element method; Galerkin method; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Methods; Partial differential equations; Polynomials; Theoretical; 65M15; Weak Galerkin finite element method; 65N15; 65N12; Second-order elliptic equation; Error estimate; Stability analysis; 65M60; 65N30
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-017-0496-6

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 ) × P j ( ∂ T ) ‖ P ℓ ( T ) 2 , where k ≥ 1 is the degree of polynomials in the interior of the element T , j ≥ 0 is the degree of polynomials on the boundary of T , and ℓ ≥ 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.