Asymptotically exact a posteriori error estimates of eigenvalues by the Crouzeix-Raviart element and enriched Crouzeix-Raviart element. (English) Zbl 1432.65168
Summary: Two asymptotically exact a posteriori error estimates are proposed for eigenvalues by the nonconforming Crouzeix-Raviart and enriched Crouzeix-Raviart elements. The main challenge in the design of such error estimators comes from the nonconformity of the finite element spaces used. Such nonconformity causes two difficulties: the first is the construction of high accuracy gradient recovery algorithms, and the second is a computable high accuracy approximation of a consistency error term. The first difficulty was solved for both nonconforming elements in a previous paper. Two methods are proposed to solve the second difficulty in the present paper. In particular, this solution allows the use of high accuracy gradient recovery techniques. Further, a postprocessing algorithm is designed by utilizing asymptotically exact a posteriori error estimators to construct the weights of a combination of two approximate eigenvalues. This algorithm requires solving only one eigenvalue problem and admits high accuracy eigenvalue approximations both theoretically and numerically.
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N25 | Numerical methods for eigenvalue problems for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
Keywords:
eigenvalue problems; nonconforming elements; asymptotically exact a posteriori error estimatesReferences:
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