The a posteriori error estimates and an adaptive algorithm of the FEM for transmission eigenvalues for anisotropic media

TL;DR

This paper develops and analyzes an a posteriori error estimator for the finite element method applied to transmission eigenvalues in anisotropic media, enabling adaptive algorithms with proven reliability and efficiency.

Contribution

It introduces a residual-based a posteriori error estimator for transmission eigenvalues in anisotropic media, with theoretical proofs of reliability and effectiveness, and demonstrates optimal convergence in numerical experiments.

Findings

Estimator is reliable for eigenfunctions and eigenvalues.

Method achieves optimal order convergence with piecewise polynomials.

Numerical results confirm efficiency and accuracy.

Abstract

The transmission eigenvalue problem arising from the inverse scattering theory is of great importance in the theory of qualitative methods and in the practical applications. In this paper, we study the transmission eigenvalue problem for anisotropic inhomogeneous media in $\Omega\subset \mathbb{R}^d$,(d=2,3). Using the T-coercivity and the spectral approximation theory, we derive an a posteriori estimator of residual type and prove its effectiveness and reliability for eigenfunctions. In addition, we also prove the reliability of the estimator for transmission eigenvalues. The numerical experiments indicate our method is efficient and can reach the optimal order $DoF^{-2m/d}$ by using piecewise polynomials of degree $m$ for real eigenvalues.