Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation |
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Authors: | Thirupathi Gudi Neela Nataraj Amiya K Pani |
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Affiliation: | (1) Center for Computation and Technology, Louisiana State University, 216 Johnston Hall, Baton Rouge, LA 70803, USA;(2) Industrial Mathematics Group, Department of Mathematics, Indian Institute of Technology, Bombay, Powai, Mumbai, 400076, India |
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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≥3, a priori error estimates of u−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−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.
Supported by DST-DAAD (PPP-05) project. |
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Keywords: | hp-Finite elements Mixed discontinuous Galerkin method Biharmonic problem Error estimates |
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