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Adaptive finite element methods for elliptic problems: abstract framework and applications. (English) Zbl 1191.65158
The authors introduce a residual type error estimator and analyze the convergence of an adaptive algorithm in a fairly general framework (continuous elliptic problems in Hilbert spaces, etc.). Two numerical examples are carried out in order to underline the performance of the estimator and the convergence of the adaptive algorithm. Then they solve a Dirichlet boundary value problem for some linear convection-diffusion-reaction problems by the discontinuous Galerkin method and introduce an a posteriori estimator.

MSC:
65N30 Finite element, Rayleigh-Ritz and Galerkin 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
35J25 Boundary value problems for second-order elliptic equations
65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs
34G10 Linear differential equations in abstract spaces
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