Preconditioning of rotated bilinear non-conforming FEM systems

Dr. Ivan Georgiev

Nov. 26, 2008, noon HF 9901

In this talk we present a new framework for multilevel preconditioning of large sparse systems of linear algebraic equations arising from the interior penalty discontinuous Galerkin approximation of second-order elliptic boundary value problems. Though the focus is on a particular family of rotated bilinear non-conforming (Rannacher-Turek) finite elements in two space dimensions (2D) the proposed rather general setting is neither limited to this particular choice of elements nor to 2D problems. One innovative contribution of this work is the construction of robust methods for problems with large jumps (several orders of magnitude) in the PDE coefficients that can only be resolved on the finest finite element mesh.

The second part of this talk will be devoted to incomplete factorization preconditioning. A locally optimized construction for an M-matrix approximation of the global stiffness matrix is the first step of the proposed algorithm. Symbolic solution technique is applied on element level for the arising local optimization problems. Then, the preconditioner is obtained by modified incomplete Cholesky factorization of the auxiliary global M-matrix. An important achievement of this work is the developed original robust preconditioning scheme for strongly anisotropic problems based on properly skewed meshes.

The talk is based on joint work with S. Margenov form IPP-BAS (Bulgaria) and with J. Kraus and J. Schicho from RICAM (Austria).