An adaptive C0IPG method for the Helmholtz transmission eigenvalue problem

TL;DR

This paper develops an adaptive $C^0$IPG method for the Helmholtz transmission eigenvalue problem, providing error indicators, proving their reliability, and demonstrating optimal convergence through numerical experiments.

Contribution

It introduces an adaptive $C^0$IPG approach with new a posteriori error indicators for eigenfunctions and eigenvalues, and proves their effectiveness.

Findings

The adaptive algorithm achieves optimal convergence rates.

Error indicators are reliable and efficient.

Numerical results confirm the method's effectiveness.

Abstract

The interior penalty methods using $C^0$ Lagrange elements ($C^0$IPG) developed in the last decade for the fourth order problems are an interesting topic in academia at present. In this paper, we discuss the adaptive fashion of $C^0$IPG method for the Helmholtz transmission eigenvalue problem.We give the a posteriori error indicators for primal and dual eigenfunctions, and prove their reliability and efficiency. We also give the a posteriori error indicator for eigenvalues and design a $C^0$IPG adaptive algorithm. Numerical experiments show that this algorithm is efficient and can get the optimal convergence rate.