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 [9], whereas a residual-type a posteriori error estimator for a quadratic C0-IPDG method applied to the biharmonic equation has been developed and analyzed in [8]. Following the convergence analysis of adaptive IPDG methods for second order elliptic problems [6], 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.
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