$$hp$$-Optimal Interior Penalty Discontinuous Galerkin Methods for the Biharmonic Problem

Abstract

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 $$\mathcal C^0$$ -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 presence of singular essential boundary conditions.