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Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method. (English) Zbl 1254.65120
An adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators is analyzed. The authors formulate the method on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension $$\geq 2.$$ It is proved that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops. The authors design a refinement procedure maintaining the level of nonconformity uniformly bounded and also prove that the approximation classes using continuous and discontinuous finite elements are equivalent. They prove that the ADFEM (and the AFEM on nonconforming meshes) yields a decay rate of energy error plus oscillation in terms of the number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.

##### MSC:
 65N12 Stability and convergence of numerical methods for boundary value problems involving PDEs 65N30 Finite element, Rayleigh-Ritz and Galerkin methods 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 65N15 Error bounds for boundary value problems involving PDEs
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