A greedy MOR method for the tracking of eigensolutions to parametric elliptic PDEs

TL;DR

This paper presents a sparse grid adaptive algorithm for tracking eigensolutions of parametric elliptic PDEs, effectively detecting eigenvalue crossings through a combination of a priori matching and a posteriori verification.

Contribution

The paper introduces a novel greedy method utilizing sparse grid refinement for eigensolution tracking in parametric elliptic PDEs, with an integrated error-driven verification process.

Findings

Numerical tests demonstrate the scheme's effectiveness.

The method accurately detects eigenvalue crossings.

The approach shows good computational performance.

Abstract

In this paper we introduce an algorithm based on a sparse grid adaptive refinement, for the approximation of the eigensolutions to parametric problems arising from elliptic partial differential equations. In particular, we are interested in detecting the crossing of the hypersurfaces describing the eigenvalues as a function of the parameters. The a priori matching is followed by an a posteriori verification, driven by a suitably defined error indicator. At a given refinement level, a sparse grid approach is adopted for the construction of the grid of the next level, by using the marking given by the a posteriori indicator. Various numerical tests confirm the good performance of the scheme.