A Multi-level Mixed Element scheme of the two dimensional helmholtz transmission eigenvalue problem

TL;DR

This paper introduces a multi-level mixed element scheme for solving the two-dimensional Helmholtz transmission eigenvalue problem on complex polygonal domains, achieving optimal convergence and computational efficiency.

Contribution

It develops a novel multi-level mixed finite element scheme for the Helmholtz transmission eigenvalue problem on non-rectangular domains, with proven optimal convergence and cost.

Findings

Achieves optimal convergence rate

Ensures optimal computational cost

Applicable to polygonal domains not covered by rectangle grids

Abstract

In this paper, we present a multi-level mixed element scheme for the Helmholtz transmission eigenvalue problem on polygonal domains that are not necessarily able to be covered by rectangle grids. We first construct an equivalent linear mixed formulation of the transmission eigenvalue problem and then discretize it with Lagrangian finite elements of low regularities. The proposed scheme admits a natural nested discretization, based on which we construct a multi-level scheme. Optimal convergence rate and optimal com- putational cost can be obtained with the scheme.