Multilevel Preconditioning of Elliptic Problems Discretized by a Class of Discontinuous Galerkin Methods

We present optimal order preconditioners for certain discontinuous Galerkin (DG) finite element discretizations of elliptic boundary value problems. A specific assembling process is proposed which allows us to use the hierarchy of geometrically nested meshes. We consider two variants of hierarchical splittings and study the angle between the resulting subspaces. Applying the corresponding two-level basis transformation recursively a sequence of algebraic problems is generated that can be associated with a hierarchy of coarse versions of DG approximations of the original problem. New bounds for the constant γ in the strengthened Cauchy-Bunyakowski-Schwarz inequality are derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach.