Generalized weak Galerkin finite element methods for second order elliptic problems

TL;DR

This paper introduces a generalized weak Galerkin finite element method for second order elliptic problems, allowing flexible polynomial choices and general partitions, with proven error estimates and supporting numerical experiments.

Contribution

It develops a new generalized weak Galerkin method with a novel discrete weak gradient, applicable to arbitrary polynomial combinations and polytopal meshes.

Findings

Error estimates in energy and L2 norms established.

Method performs well on various polytopal partitions.

Numerical experiments confirm theoretical results.

Abstract

This article proposes and analyzes the generalized weak Galerkin ({\rm g}WG) finite element method for the second order elliptic problem. A generalized discrete weak gradient operator is introduced in the weak Galerkin framework so that the {\rm g}WG methods would not only allow arbitrary combinations of piecewise polynomials defined in the interior and on the boundary of each local finite element, but also work on general polytopal partitions. Error estimates are established for the corresponding numerical functions in the energy norm and the usual $L^2$ norm. A series of numerical experiments are presented to demonstrate the performance of the newly proposed {\rm g}WG method.