Weak Galerkin methods for elliptic interface problems on curved polygonal partitions

TL;DR

This paper introduces a weak Galerkin method for elliptic interface problems on curved polygonal partitions, transforming complex interface conditions into simpler boundary conditions and achieving optimal error estimates.

Contribution

The paper develops a novel WG method that handles curved interfaces by converting jump conditions into Dirichlet conditions, with proven optimal error estimates.

Findings

Optimal error estimates in discrete $H^1$ and $L^2$ norms.

Numerical results demonstrating robustness and effectiveness.

Method simplifies analysis of complex interface problems.

Abstract

This paper presents a new weak Galerkin (WG) method for elliptic interface problems on general curved polygonal partitions. The method's key innovation lies in its ability to transform the complex interface jump condition into a more manageable Dirichlet boundary condition, simplifying the theoretical analysis significantly. The numerical scheme is designed by using locally constructed weak gradient on the curved polygonal partitions. We establish error estimates of optimal order for the numerical approximation in both discrete $H^1$ and $L^2$ norms. Additionally, we present various numerical results that serve to illustrate the robust numerical performance of the proposed WG interface method.