Error Analysis of Symmetric Linear/Bilinear Partially Penalized Immersed Finite Element Methods for Helmholtz Interface Problems

TL;DR

This paper provides an error analysis for symmetric linear and bilinear partially penalized immersed finite element methods applied to Helmholtz interface problems, establishing optimal error bounds under certain regularity assumptions.

Contribution

It introduces an error analysis framework for PPIFE methods for Helmholtz interface problems, deriving optimal error bounds in energy and L2 norms.

Findings

Optimal error bounds established for PPIFE solutions

Numerical validation confirms theoretical error estimates

Analysis assumes piecewise H^2 regularity of solutions

Abstract

This article presents an error analysis of the symmetric linear/bilinear partially penalized immersed finite element (PPIFE) methods for interface problems of Helmholtz equations. Under the assumption that the exact solution possesses a usual piecewise $H^2$ regularity, the optimal error bounds for the PPIFE solutions are derived in an energy norm and the usual $L^2$ norm provided that the mesh size is sufficiently small. A numerical example is conducted to validate the theoretical conclusions.