Curved Elements in Weak Galerkin Finite Element Methods

TL;DR

This paper provides a mathematical analysis of weak Galerkin finite element methods for the Poisson equation involving curved elements, establishing optimal error estimates and demonstrating effectiveness through numerical results.

Contribution

It introduces a rigorous analysis of weak Galerkin methods with curved elements, including error estimates and validation on curved polygonal partitions.

Findings

Optimal error estimates in $H^1$-norm and $L^2$-norm.

Numerical results confirm theoretical error bounds.

Effective application to curved polygonal partitions.

Abstract

A mathematical analysis is established for the weak Galerkin finite element methods for the Poisson equation with Dirichlet boundary value when the curved elements are involved on the interior edges of the finite element partition or/and on the boundary of the whole domain in two dimensions. The optimal orders of error estimates for the weak Galerkin approximations in both the $H^1$-norm and the $L^2$-norm are established. Numerical results are reported to demonstrate the performance of the weak Galerkin methods on general curved polygonal partitions.