A Priori Error Estimates for Some Discontinuous Galerkin Immersed Finite Element Methods

TL;DR

This paper derives optimal a priori error estimates for a class of discontinuous Galerkin immersed finite element methods applied to elliptic interface problems, demonstrating their convergence and local mesh refinement capabilities.

Contribution

It introduces a priori error estimates for DG-IFE methods, highlighting their optimal convergence and suitability for local mesh refinement in interface problems.

Findings

Methods converge optimally in energy norm

Numerical results confirm convergence and refinement

DG-IFE methods effectively handle interface problems

Abstract

In this paper, we derive a priori error estimates for a class of interior penalty discontinuous Galerkin (DG) methods using immersed finite element (IFE) functions for a classic second-order elliptic interface problem. The error estimation shows that these methods can converge optimally in a mesh-dependent energy norm. The combination of IFEs and DG formulation in these methods allows local mesh refinement in the Cartesian mesh structure for interface problems. Numerical results are provided to demonstrate the convergence and local mesh refinement features of these DG-IFE methods.