Two-sided bounds of eigenvalues - local efficiency and convergence of adaptive algorithm

TL;DR

This paper extends a method for computing lower bounds of eigenvalues in symmetric elliptic problems, introducing local flux reconstruction, proving estimator efficiency and convergence, and demonstrating practical effectiveness through numerical examples.

Contribution

It generalizes the eigenvalue lower bound method to higher eigenvalues, introduces a local flux reconstruction approach, and proves the efficiency and convergence of the adaptive algorithm.

Findings

The method provides reliable lower bounds for higher eigenvalues.

The local flux reconstruction enables efficient implementation.

The adaptive algorithm converges and performs well in numerical tests.

Abstract

We generalize and analyse the method for computing lower bounds of the principal eigenvalue proposed in our previous paper (I. Sebestova, T. Vejchodsky, SIAM J. Numer. Anal. 2014). This method is suitable for symmetric elliptic eigenvalue problems with mixed boundary conditions of Dirichlet, Neumann, and Robin type and it is based on a posteriori error analysis using flux reconstructions. We improve the original result in several aspects. We show how to obtain lower bounds even for higher eigenvalues. We present a local approach for the flux reconstruction enabling efficient implementation. We prove the equivalence of the resulting estimator with the classical residual estimator and consequently its local efficiency. We also prove the convergence of the corresponding adaptive algorithm. Finally, we illustrate the practical performance of the method by numerical examples.