A priori and a posteriori error analysis for the mixed discontinuous Galerkin finite element approximations of the biharmonic problems

In this article, a new mixed discontinuous Galerkin finite element method is proposed for the biharmonic equation in two or three-dimension space. It is amenable to an efficient implementation displaying new convergence properties. Through an auxiliary variable p = -Δu, we rewrite the problem into a two-order system. Then, the a priori error estimates are derived in L² norm and in the broken DG norm for both u and p. We prove that, when polynomials of degree r (r ≥ 1) are used, we obtain the optimal convergence rate of order r + 1 in L² norm and of order r in DG norm for u, and the order r in both norms for p = -Δu. The numerical experiments illustrate the theoretic order of convergence. For the purpose of adaptive finite element method, the a posteriori error estimators are also proposed and proved to field a sharp upper bound. We also provide numerical evidence that the error estimators and indicators can effectively drive the adaptive strategies.