A Subspace Correction Method for Discontinuous Galerkin Discretizations of Linear Elasticity Equations

We study preconditioning techniques for discontinuous Galerkin discretizations of linear elasticity problems in primal (displacement) formulation. We propose a space splitting which gives rise to uniform preconditioners for the Interior Penalty (IP) Finite Element (FE) discretizations recently introduced in the works [12, 18, 19]. For the case when Dirichlet boundary conditions are imposed on the entire boundary, the action of the preconditioner is equivalent to solving several (2 or 3) Laplace equations and the condition number of the preconditioned system is uniformly bounded with respect to both the Poisson ratio and the mesh size. However, when natural (i.e. traction free) boundary conditions are prescribed on part of the boundary the situation is much more subtle, and we present here a preconditioning technique which reduces the solution of the linear algebraic system corresponding to the IP Galerkin method to a solution of a discretization with nonconforming Crouzeix-Raviart elements. There are several works on preconditioning discretized linear elasticity equations for conforming or non-conforming finite element methods [5, 11, 14, 15, 16]. However, to our knowledge the works related to the preconditioning of the Discontinuous Galerkin (DG) discretizations of linear elasticity equations are very limited. Our work is focused on preconditioning a particular type of DG methods namely interior penalty methods, and we construct a uniform preconditioner for the Symmetric Interior Penalty Galerkin (SIPG) discretization. The main ingredient is a natural splitting of the DG space. Such a splitting was introduced in [4] in the context of designing subspace correction methods and also considered in [8] in a different context. In [4] it was shown that subspace correction methods for a discretization of scalar elliptic equations, based on such a natural splitting of the DG space lead to uniform preconditioners for the symmetrized DG schemes and to uniformly convergent iterative methods for the nonsymmetric DG schemes. Here we have also extended some of the results from [4] to vector field problems, including results for Nonsymmetric Interior Penalty (NIPG) and Incomplete Interior Penalty (IIPG) discretizations. The rest of the paper is organized as follows. We introduce the problem and the basic notation in §2. Next, in §3 we introduce the corresponding DG discretizations and recall some of their stability and approximation properties. In the last section §4, we introduce the subspace correction methods, and we prove that they give rise to a uniform preconditioner of the symmetric IP method.