Guaranteed eigenvalue and eigenvector bounds

eigs_conf_str

First eigenvalue and eigenvector bounds on a square (structured mesh, 1st order conforming finite elements)

eigs_DG_adapt

First eigenvector bounds on an L-shaped domain (adaptively refined mesh, 1st and 2nd order discontinuous Galerkin elements)

eigs_hole_mesh

Adaptively refined mesh, domain with a hole

eigs_hole_incl

Guaranteed inclusion of the first eigenvalue, domain with a hole

clusters

Eigenvalue and eigenvector bounds, 2-cluster on a square (left) and 3-cluster on an L-shaped domain (right), 1st order conforming finite elements

Main results:

  • guaranteed upper and lower bounds for an arbitrary simple eigenvalue, multiple eigenvalue, or a cluster of eigenvalues;
  • guaranteed energy error bounds on the associated eigenvector(s);
  • efficiency (optimal convergence rate) and polynomial-degree robustness;
  • multiplicative constant tending to one under an elliptic regularity assumption on the corresponding
    source problem;
  • inexact algebraic solvers taken into account;
  • unified framework covering basically any numerical method (conforming, nonconforming, discontinuous Galerkin, and mixed finite elements);
  • abstract framework for any second-order self-adjoint elliptic linear operator with compact resolvent.

Details:

  • with Eric Cancès, Geneviève Dusson, Yvon Maday, and Benjamin Stamm;
  • conforming setting, simple eigenvalue: paper
  • unified framework, simple eigenvalue: paper and preprint, presentation
  • conforming setting, multiple eigenvalues and clusters: preprint, presentation

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