Penalty-Free Any-Order Weak Galerkin FEMs for Elliptic Problems on Quadrilateral Meshes

This paper presents a family of weak Galerkin finite element methods for elliptic boundary value problems on convex quadrilateral meshes. These new methods use degree k≥0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$k \ge 0$$\end{document} polynomials separately in element interiors and on edges for approximating the primal variable. The discrete weak gradients of these shape functions are established in the local Arbogast–Correa ACk\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$AC_k $$\end{document} spaces. These discrete weak gradients are then used to approximate the classical gradient in the variational formulation. These new methods do not use any nonphysical penalty factor but produce optimal-order approximation to the primal variable, flux, normal flux, and divergence of flux. Moreover, these new solvers are locally conservative and offer continuous normal fluxes. Numerical experiments are presented to demonstrate the accuracy of this family of new methods.