The Compact Discontinuous Galerkin Method for Nearly Incompressible Linear Elasticity

A compact discontinuous Galerkin method (CDG) is devised for nearly incompressible linear elasticity, through replacing the global lifting operator for determining the numerical trace of stress tensor in a local discontinuous Galerkin method (cf. Chen et al., Math Probl Eng 20, 2010) by the local lifting operator and removing some jumping terms. It possesses the compact stencil, that means the degrees of freedom in one element are only connected to those in the immediate neighboring elements. Optimal error estimates in broken energy norm, $H^1$ -norm and $L^2$ -norm are derived for the method, which are uniform with respect to the Lamé constant $\lambda .$ Furthermore, we obtain a post-processed $H(\text{ div})$ -conforming displacement by projecting the displacement and corresponding trace of the CDG method into the Raviart–Thomas element space, and obtain optimal error estimates of this numerical solution in $H(\text{ div})$ -seminorm and $L^2$ -norm, which are uniform with respect to $\lambda .$ A series of numerical results are offered to illustrate the numerical performance of our method.