Two‐level preconditioning of discontinuous Galerkin approximations of second‐order elliptic equations

This paper reviews some known and proposes some new preconditioning methods for a number of discontinuous Galerkin (or DG) finite element approximations for elliptic problems of second order. Nested hierarchy of meshes is generally assumed. Our approach utilizes a general two‐level scheme, where the finite element space for the DG method is decomposed into a subspace (viewed as an auxiliary or ‘coarse’ space), plus a correction which can be handled by a standard smoothing procedure. We consider three different auxiliary subspaces, namely, piecewise linear C 0 ‐conforming functions, piecewise linear functions that are continuous at the centroids of the edges/faces (Crouzeix‐Raviart finite elements) and piecewise constant functions over the finite elements. To support the theoretical results, we also present numerical experiments for 3‐D model problem showing uniform convergence of the constructed methods.