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Mixed discontinuous Galerkin finite element method for the biharmonic equation. (English) Zbl 1203.65254
Summary: We first split the biharmonic equation Δ 2 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 p3, a priori error estimates of u-u h in the broken H 1 norm as well as in the L 2 norm which are optimal in h and suboptimal in p are derived. Moreover, an 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 method also converges, but with suboptimal convergence rate. Finally, numerical experiments are presented to illustrate the theoretical results.
MSC:
65N30Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (BVP of PDE)
65N15Error bounds (BVP of PDE)
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