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Adaptive finite element methods for the Laplace eigenvalue problem. (English) Zbl 1222.65122

The authors present an adaptive finite element method for the Laplace eigenvalue problem on bounded polygonal or polyhedral domains based on conforming P1 elements. They consider a residual type a posteriori error estimator consisting of element and edge residuals. The a posteriori error analysis also involves an oscillation term. The refinement of the triangulation is done by a bulk criterion (Dörfler marking) and the refinement strategy relies on repeated bisection. The convergence result states a reduction both in the energy norm of the error and in the oscillation term. Numerical experiments are included to illustrate the performance of the adaptive algorithm.

MSC:

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
65N15 Error bounds for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35P15 Estimates of eigenvalues in context of PDEs
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References:

[1] Grisvard, SIAM Elliptic Problems in Nonsmooth Domains Pitman Boston Grubisic and On estimators for eigenvalue / eigenvector approximations, Numer Anal 47 pp 1067– (2009)
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