Higher order stable generalized finite element method for the elliptic eigenvalue problem with an interface in 1D

TL;DR

This paper extends higher order stable generalized finite element methods to 1D elliptic eigenvalue problems with interfaces, providing optimal error estimates and validating them through numerical experiments.

Contribution

It generalizes SGFEM to arbitrary order elements and establishes optimal convergence for eigenvalue problems with interfaces.

Findings

Optimal error convergence for approximate solutions

Validated theoretical error estimates with numerical examples

Extended SGFEM to higher order elements for interface problems

Abstract

We study the generalized finite element methods (GFEMs) for the second-order elliptic eigenvalue problem with an interface in 1D. The linear stable generalized finite element methods (SGFEM) were recently developed for the elliptic source problem with interfaces. We first generalize SGFEM to arbitrary order elements and establish the optimal error convergence of the approximate solutions for the elliptic source problem with an interface. We then apply the abstract theory of spectral approximation of compact operators to establish the error estimation for the eigenvalue problem with an interface. The error estimations on eigenpairs strongly depend on the estimation of the discrete solution operator for the source problem. We verify our theoretical findings in various numerical examples including both source and eigenvalue problems.