hp-optimal interior penalty discontinuous Galerkin methods for the biharmonic problem
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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 presence of singular essential boundary conditions.
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