Convergence and optimality of an adaptive modified weak Galerkin finite element method

Abstract

An adaptive modified weak Galerkin method (AmWG) for an elliptic problem is studied in this paper, in addition to its convergence and optimality. The modified weak Galerkin bilinear form is simplified without the need of the skeletal variable, and the approximation space is chosen as the discontinuous polynomial space as in the discontinuous Galerkin method. Upon a reliable residual-based a posteriori error estimator, an adaptive algorithm is proposed together with its convergence and quasi-optimality proved for the lowest order case. The primary tool is to bridge the connection between the modified weak Galerkin method and the Crouzeix-Raviart nonconforming finite element. Unlike the traditional convergence analysis for methods with a discontinuous polynomial approximation space, the convergence of AmWG is penalty parameter free. Numerical results are presented to support the theoretical results.

Extended reference to include recent contributions than the published version

References in corpus (4)

• A Computational Study of the Weak Galerkin Method for Second-Order Elliptic Equations
• A Numerical Study on the Weak Galerkin Method for the Helmholtz Equation
• A C^0-Weak Galerkin Finite Element Method for the Biharmonic Equation
• A Recovery-Based A Posteriori Error Estimator for H(curl) Interface Problems