Guaranteed eigenvalue and eigenvector bounds


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


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


Adaptively refined mesh, domain with a hole


Guaranteed inclusion of the first eigenvalue, domain with a hole


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.


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