hp-Version a priori Error Analysis of Interior Penalty Discontinuous Galerkin Finite Element Approximations to the Biharmonic Equation

Authors:
Igor Mozolevski;Endre Süli;Paulo R. Bösing
Affiliations:
Mathematics Department, Federal University of Santa Catarina, Trindade, Brazil SC- 88040-900;Computing Laboratory, University of Oxford, Oxford, UK OX1 3QD;Applied Mathematics Department, IME, University of São Paulo, São Paulo, Brazil SP- 05508-090
Venue:
Journal of Scientific Computing
Year:
2007

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Abstract

We consider the symmetric formulation of the interior penalty discontinuous Galerkin finite element method for the numerical solution of the biharmonic equation with Dirichlet boundary conditions in a bounded polyhedral domain in $$\mathbb{R}^d, d \geqslant 2$$. For a shape-regular family of meshes consisting of parallelepipeds, we derive hp-version a priori bounds on the global error measured in the L2 norm and in broken Sobolev norms. Using these, we obtain hp-version bounds on the error in linear functionals of the solution. The bounds are optimal with respect to the mesh size h and suboptimal with respect to the degree of the piecewise polynomial approximation p. The theoretical results are confirmed by numerical experiments, and some practical applications in Poisson---Kirchhoff thin plate theory are presented.