hp-optimal interior penalty discontinuous Galerkin methods for the biharmonic problem

Overview & Synopsis

We prove hp-optimal error estimates for interior penalty discontinuous Galerkin methods (IPDG) for the biharmonic problem with homogeneous essential boundary conditions. We consider tensor product-type meshes in two and three dimensions and triangular meshes in two dimensions. An essential ingredient in the analysis is the construction of a global H 2 piecewise polynomial approximants with hp-optimal approximation properties over the given meshes. The hp-optimality is also discussed for C 0-IPDG in two and three dimensions and the stream formulation of the Stokes problem in two dimensions. Numerical experiments validate the theoretical predictions and reveal that p-suboptimality occurs in the presence of singular essential boundary conditions.