Partially Penalized Immersed Finite Element Methods for Elliptic Interface Problems

TL;DR

This paper introduces stabilized immersed finite element methods with penalty terms for elliptic interface problems, achieving optimal convergence and robustness on structured meshes with complex interfaces.

Contribution

The paper develops new stabilized IFE methods with penalty terms that improve convergence and stability for elliptic interface problems on structured meshes.

Findings

Optimal convergence in H1-norm for the proposed methods.

Convergence rates remain stable with mesh refinement.

Trace inequalities established for IFE functions.

Abstract

This article presents new immersed finite element (IFE) methods for solving the popular second order elliptic interface problems on structured Cartesian meshes even if the involved interfaces have nontrivial geometries. These IFE methods contain extra stabilization terms introduced only at interface edges for penalizing the discontinuity in IFE functions. With the enhanced stability due to the added penalty, not only these IFE methods can be proven to have the optimal convergence rate in the H1-norm provided that the exact solution has sufficient regularity, but also numerical results indicate that their convergence rates in both the H1-norm and the L2-norm do not deteriorate when the mesh becomes finer which is a shortcoming of the classic IFE methods in some situations. Trace inequalities are established for both linear and bilinear IFE functions that are not only critical for the error analysis of these new IFE methods, but also are of a great potential to be useful in error analysis for other IFE methods.