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

Abstract

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 , 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 () 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 . 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. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 318–353, 2017