A simple low-degree optimal finite element scheme for the elastic transmission eigenvalue problem

TL;DR

This paper introduces a straightforward finite element method using cubic polynomials for elastic transmission eigenvalues, achieving optimal convergence without extra stabilization, simplifying implementation compared to existing schemes.

Contribution

It proposes a low-degree, stable finite element scheme for elastic transmission eigenvalues that is easy to implement and achieves optimal convergence.

Findings

Achieves optimal convergence rate with cubic polynomials.

Does not require extra stabilization, simplifying implementation.

Outperforms other low-degree, nonconforming schemes.

Abstract

The paper presents a finite element scheme for the elastic transmission eigenvalue problem written as a fourth order eigenvalue problem. The scheme uses piecewise cubic polynomials and obtains optimal convergence rate. Compared with other low-degree and nonconforming finite element schemes, the scheme inherits the continuous bilinear form which does not need extra stabilizations and is thus simple to implement.