Convergence analysis of an adaptive interior penalty discontinuous Galerkin method for the biharmonic problem

For the biharmonic problem, we study the convergence of adaptive C0-Interior Penalty Discontinuous Galerkin (C0-IPDG) methods of any polynomial order. We note that C0-IPDG methods for fourth order elliptic boundary value problems have been suggested in several publications, whereas residual-type a posteriori error estimators for C0-IPDG methods applied to the biharmonic equation have been developed and analyzed in [8, 18]. Following the convergence analysis of adaptive IPDG methods for second order elliptic problems, we prove a contraction property for a weighted sum of the C0-IPDG energy norm of the global discretization error and the estimator. The proof of the contraction property is based on the reliability of the estimator, a quasi-orthogonality result, and an estimator reduction property. Numerical results are given that illustrate the performance of the adaptive C0-IPDG approach.

Journal of Numerical Mathematics