Efficient Preconditioning for the Discontinuous Galerkin Finite Element Method by Low-Order Elements

Abstract

We derive and analyze a block diagonal preconditioner for the linear problems arising from a discontinuous Galerkin finite element discretization. The method can be applied to second-order self-adjoint elliptic boundary value problems and exploits the natural decomposition of the discrete function space into a global low-order subsystem and components of higher polynomial degree. Similar to results for the p-version of the conforming FEM, we prove for the interior penalty discontinuous Galerkin discretization that the condition number of the preconditioned system grows as p² (log p + 1)², where p is the polynomial degree of the discrete function space. Numerical experiments demonstrate the performance of the method.