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

## Dr. Satyendra K. Tomar

Jan. 30, 2007, 3:30 p.m. T 1010

In this talk we shall present our work on devising optimal order multilevel preconditioners for interior-penalty discontinuous Galerkin (DG) finite element discretizations of three-dimensional (3D) anisotropic elliptic boundary-value problems.

In this work 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 $\gamma$ in the strengthened Cauchy-Bunyakowski-Schwarz inequality is derived. Several numerical results, which support the theoretical analysis and demonstrate the potential of this approach, will be presented.