A boundary-corrected weak Galerkin mixed finite method for elliptic interface problems with curved interfaces

TL;DR

This paper introduces a boundary-corrected weak Galerkin mixed finite element method for elliptic interface problems with curved interfaces, improving accuracy and reducing complexity on polygonal meshes.

Contribution

It presents a novel boundary correction technique using Taylor expansion to handle curved interfaces without complex curved element integration.

Findings

Achieves optimal-order convergence in energy norm

Eliminates the need for curved element numerical integration

Supports arbitrary-order discretizations

Abstract

We propose a boundary-corrected weak Galerkin mixed finite element method for solving elliptic interface problems in 2D domains with curved interfaces. The method is formulated on body-fitted polygonal meshes, where interface edges are straight and may not align exactly with the curved physical interface. To address this discrepancy, a boundary value correction technique is employed to transfer the interface conditions from the physical interface to the approximate interface using a Taylor expansion approach. The Neumann interface condition is then weakly imposed in the variational formulation. This approach eliminates the need for numerical integration on curved elements, thereby reducing implementation complexity. We establish optimal-order convergence in the energy norm for arbitrary-order discretizations. Numerical results are provided to support the theoretical findings.