Mixed Discontinuous Galerkin Finite Element Method for the Biharmonic Equation

In this paper, we first split the biharmonic equation Δ2u=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 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 H1 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.