Mixed Discontinuous Galerkin Finite Element Method For The Biharmonic Equation

摘要

In this paper, we first split the biharmonic equation △~2u = f with nonhomoge-neous 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 v_h of v can easily be eliminated to reduce the discrete problem to a Schur complement system in v_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.