Discontinuous Galerkin Methods With Interior Penalty For Linear Elasticity

Linear elasticity problem mainly study the changes of stress, strain and displacement of elastic body caused by external factors such as external force or temperature changes. Since the practical problems frequently occuring in the fields of machinery, architecture, chemical industry and aerospace engineering mostly belong to such case, the researches on its numerical theories have important theoretical significance and practical value. In this work, a family of primal discontinuous Galerkin methods with interior penalty are introduced to solve the linear elasticity problem. We provide a unified framework for the analysis of these methods and show the posedness of these schemes, the optimal convergence rates in energy norm for all the methods and the optimal convergence rates in L2norm only for symmetric methods. For nonsymmetric methods, superpenalty is used to reduce adjoint inconsistency error to the point where the optimal convergence rates in L2norm are achieved. Furthermore, we develop an improved posteriori error estimate based on the Helmholtz decomposition of tensor fields. After proving its effectiveness, an adaptive discontinuous Galerkin method based on this posteriori error is proposed and some numerical examples confirming the theoretical results are presented.