Session B6 - Multiresolution and Adaptivity in Numerical PDEs
July 14, 14:30 ~ 14:55
Guaranteed and robust a posteriori bounds for Laplace eigenvalues and eigenvectors
Inria Paris, France - firstname.lastname@example.org
We derive a posteriori error estimates for numerical approximation of the Laplace eigenvalue problem with a homogeneous Dirichlet boundary condition. In particular, upper and lower bounds for an arbitrary simple eigenvalue are given. These bounds are guaranteed, fully computable, and converge with optimal speed to the given exact eigenvalue, under a separation condition from the surrounding smaller and larger exact eigenvalues that we can check in practice. Guaranteed, fully computable, optimally convergent, and polynomial-degree robust bounds on the energy error in the approximation of the associated eigenvector are derived as well, under the same hypotheses. Remarkably, there appears no unknown (solution-, regularity-, or polynomial-degree-dependent) constant in our theory. Inexact algebraic solvers are taken into account, so that the estimates are valid on each iteration and can serve for design of adaptive stopping criteria for eigenvalue solvers. The framework can be applied to conforming, nonconforming, discontinuous Galerkin, and mixed finite element approximations of arbitrary polynomial degree. We illustrate it numerically on a set of test problems.
Joint work with Eric Cances (Ecole des Ponts, France), Genevieve Dusson (University Paris 6, France), Yvon Maday (University Paris 6, France) and Benjamin Stamm (RWTH Aachen, Germany).