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Output bounds for reduced-basis approximations of symmetric positive definite eigenvalue problems. (English. Abridged French version) Zbl 0960.65063
Summary: We propose a new reduced-basis output bound method for the symmetric eigenvalue problem. The numerical procedure consists of two stages: the pre-processing stage, in which the reduced basis and associated functions are computed – “off-line” – at a prescribed set of points in parameter space; and the real-time stage, in which the approximate output of interest and corresponding rigorous error bounds are computed – “on-line” – for any new parameter value of interest. The real time calculation is very inexpensive as it requires only the solution or evaluation of very small systems. We introduce the procedure; prove the asymptotic bounding properties and optimal convergence rate of the error estimator; discuss computational considerations; and, finally, present corroborating numerical results.

MSC:
65J10 Numerical solutions to equations with linear operators (do not use 65Fxx)
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
35P15 Estimates of eigenvalues in context of PDEs
47A75 Eigenvalue problems for linear operators
65N25 Numerical methods for eigenvalue problems for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
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