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A unified analysis of quasi-optimal convergence for adaptive mixed finite element methods. (English) Zbl 1387.65118

Summary: In this paper, we present a unified analysis of both convergence and optimality of adaptive mixed finite element methods for a class of problems when the finite element spaces and corresponding a posteriori error estimates under consideration satisfy five hypotheses. We prove that these five conditions are sufficient for convergence and optimality of the adaptive algorithms under consideration. The main ingredient for the analysis is a new method to analyze both discrete reliability and quasi-orthogonality. This new method arises from an appropriate and natural choice of the norms for both the discrete displacement and stress spaces, namely, a mesh-dependent discrete \(H^1\) norm for the former and an \(L^2\) norm for the latter, and a newly defined projection operator from the discrete stress space on the coarser mesh onto the discrete divergence free space on the finer mesh. As applications, we prove these five hypotheses for the Raviart-Thomas and Brezzi-Douglas-Marini elements of the Poisson and Stokes problems in both two and three dimensions.

MSC:

65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65Y20 Complexity and performance of numerical algorithms
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References:

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