A Multilevel Correction Method for Interior Transmission Eigenvalue Problem

TL;DR

This paper introduces a multilevel correction finite element method that efficiently solves the interior transmission eigenvalue problem by reducing it to simpler linear problems and low-dimensional eigenvalue computations.

Contribution

It proposes a novel multilevel correction approach that enhances computational efficiency for transmission eigenvalue problems compared to traditional methods.

Findings

The method reduces computational cost significantly.

Numerical examples validate the theoretical efficiency gains.

The approach accurately computes transmission eigenvalues in finite element spaces.

Abstract

In this paper, we give a numerical analysis for the transmission eigenvalue problem by the finite element method. A type of multilevel correction method is proposed to solve the transmission eigenvalue problem. The multilevel correction method can transform the transmission eigenvalue solving in the finest finite element space to a sequence of linear problems and some transmission eigenvalue solving in a very low dimensional spaces. Since the main computational work is to solve the sequence of linear problems, the multilevel correction method improves the overfull efficiency of the transmission eigenvalue solving. Some numerical examples are provided to validate the theoretical results and the efficiency of the proposed numerical scheme.