Weak Galerkin finite element method for second order problems on curvilinear polytopal meshes with Lipschitz continuous edges or faces

TL;DR

This paper introduces a new weak Galerkin finite element method with specialized basis functions for solving second order elliptic problems on curved, complex meshes, achieving optimal convergence and high accuracy.

Contribution

It develops novel basis functions for curved faces in the weak Galerkin method and analyzes their effectiveness on complex, Lipschitz continuous geometries.

Findings

Achieves optimal $H^1$ and $L^2$ error convergence rates.

Supports arbitrary high-order accuracy for smooth solutions.

Numerical tests confirm theoretical error estimates.

Abstract

In this paper, we propose new basis functions defined on curved sides or faces of curvilinear elements (polygons or polyhedrons with curved sides or faces) for the weak Galerkin finite element method. Those basis functions are constructed by collecting linearly independent traces of polynomials on the curved sides/faces. We then analyze the modified weak Galerkin method for the elliptic equation and the interface problem on curvilinear polytopal meshes with Lipschitz continuous edges or faces. The method is designed to deal with less smooth complex boundaries or interfaces. Optimal convergence rates for $H^1$ and $L^2$ errors are obtained, and arbitrary high orders can be achieved for sufficiently smooth solutions. The numerical algorithm is discussed and tests are provided to verify theoretical findings.