Quasi-Optimal Convergence Rate of an Adaptive Discontinuous Galerkin Method

Abstract

We analyze an adaptive discontinuous finite element method (ADFEM) for symmetric second order linear elliptic operators. The method is formulated on nonconforming meshes made of simplices or quadrilaterals, with any polynomial degree and in any dimension $\geq2$. We prove that the ADFEM is a contraction for the sum of the energy error and the scaled error estimator between two consecutive adaptive loops. We design a refinement procedure that maintains the level of nonconformity uniformly bounded and prove that the approximation classes using continuous and discontinuous finite elements are equivalent. The geometric decay and the equivalence of classes are instrumental in deriving the optimal cardinality of the ADFEM. We show 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.