Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

  • Authors:

  • Thirupathi Gudi;Neela Nataraj;Amiya K. Pani

  • Affiliations:

  • Center for Computation and Technology, Louisiana State University, Baton Rouge, USA 70803;Industrial Mathematics Group, Department of Mathematics, Indian Institute of Technology, Mumbai, India 400076;Industrial Mathematics Group, Department of Mathematics, Indian Institute of Technology, Mumbai, India 400076

  • Venue:
  • Journal of Scientific Computing
  • Year:
  • 2008

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Abstract

In this paper, we first split the biharmonic equation Δ2 u=f with nonhomogeneous essential boundary conditions into a system of two second order equations by introducing an auxiliary variable v=Δu and then apply an hp-mixed discontinuous Galerkin method to the resulting system. The unknown approximation v h of v can easily be eliminated to reduce the discrete problem to a Schur complement system in u h , which is an approximation of u. A direct approximation v h of v can be obtained from the approximation u h of u. Using piecewise polynomials of degree p驴3, a priori error estimates of u驴u h in the broken H 1 norm as well as in L 2 norm which are optimal in h and suboptimal in p are derived. Moreover, a priori error bound for v驴v h in L 2 norm which is suboptimal in h and p is also discussed. When p=2, the preset method also converges, but with suboptimal convergence rate. Finally, numerical experiments are presented to illustrate the theoretical results.