A multilevel method for discontinuous Galerkin approximation of three-dimensional anisotropic elliptic problems

A. We construct optimal order multilevel preconditioners for interior-penalty discontinuous Galerkin (DG) finite element discretizations of three-dimensional (3D) anisotropic elliptic boundary-value problems. In this paper we extend the analysis of our approach, introduced earlier for 2D problems [20], to cover 3D problems. A specific assembling process is proposed which allows us to characterize the hierarchical splitting locally. This is also the key for a local analysis of the angle between the resulting subspaces. Applying the corresponding two-level basis transformation recursively, a sequence of algebraic problems is generated. These discrete problems can be associated with coarse versions of DG approximations (of the solution to the original variational problem) on a hierarchy of geometrically nested meshes. A new bound for the constant γ in the strengthened CauchyBunyakowski-Schwarz inequality is derived. The presented numerical results support the theoretical analysis and demonstrate the potential of this approach.