Convergence and optimality of the adaptive Morley element method. (English) Zbl 1255.65212
An analysis of the convergence and optimality of the adaptive version of the Morley element method for a 4th-order elliptic problem is carried out. Whereas adaptive conforming finite element methods applied to second order elliptic problems have been extensively studied, very few results exist for nonconforming methods in fourth order problems, the main difficulty being the nonconformity of the discrete space and the lack of the Galerkin orthogonality. The analysis here is based on a quasi-orthogonality result which holds specifically for the Morley element method and a new parameter dependent estimator. With the additional help of the discrete reliability of the estimator, it is possible to show convergence and optimality of the adaptive algorithm based on the Morley element method. The generalization to the nonconforming linear elements in two and three dimensions is also discussed.
Reviewer: Fernando Casas (Castellon)
MSC:
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65N15 | Error bounds for boundary value problems involving PDEs |
| 35J30 | Higher-order elliptic equations |
| 65N12 | Stability and convergence of numerical methods for boundary value problems involving PDEs |
Keywords:
Morley element method; fourth-order elliptic problems; adaptivity; convergence; optimality; finite element; nonconforming method; algorithmReferences:
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