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Reduced basis approximation and a posteriori error estimates for parametrized elliptic eigenvalue problems. (English) Zbl 1355.65149

The rough contents of this large and at the same time dense article are as follows: 1. Introduction, 2. Parametrized elliptic eigenvalue problems, 3. The reduced basis approximation, 4. A posteriori error estimates, 5. Numerical results and 6. Conclusions. An appendix, devoted to an extension of the Baur-Fike theorem to generalized eigenvalue problems for two symmetric positive definite matrices, and references (37 entries) are also available. First, the authors introduce the family of parametrized elliptic eigenvalue problems of interest along with its high-fidelity Galerkin-finite element approximation for the first eigenpair. Then, they formulate the reduced basis (RB) approximation along with a greedy algorithm for the efficient assembling of reduced basis spaces. As their main result, based on the dual weighted residual theory, they establish a posteriori error estimates for the parametrized eigenvalue problem and prove their reliability. Several numerical results, related to both affinely and non-affinely parametrized eigenproblems, are carried out. They validate the newly introduced RB method for fast and robust approximation of parametrized elliptic eigenvalue problems.

MSC:

65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs

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