Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

In this paper, we first split the biharmonic equation !2u = f with nonhomogeneous essential boundary conditions into a system of two second order equations by introducing an auxiliary variable v = !u and then apply an hp-mixed discontinuous Galerkin method to the resulting system. The unknown approximation vh of v can easily be eliminated to reduce the discrete problem to a Schur complement system in uh, which is an approximation of u. A direct approximation vh of v can be obtained from the approximation uh of u. Using piecewise polynomials of degree p ≥ 3, a priori error estimates of u − uh in the broken H 1 norm as well as in L2 norm which are optimal in h and suboptimal in p are derived. Moreover, a priori error bound for v − vh in L2 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.