Krylov-Subspace Preconditioners for Discontinuous Galerkin Finite Element Methods

Krylov-Subspace Preconditioners for Discontinuous Galerkin Finite Element Methods

Georgoulis, E.H.
Loghin, D.

2006-09-05

Standard (conforming) finite element approximations of convection-dominated convection-diffusion problems often exhibit poor stability properties that manifest themselves as non-physical oscillations polluting the numerical solution. Various techniques have been proposed for the stabilisation of finite element methods (FEMs) for convection-diffusion problems, such as the popular streamline upwind Petrov-Galerkin (SUPG) method, and its variants. During the last decade, families of discontinuous Galerkin finite element methods (DGFEMs) have been proposed for the numerical solution of convection-diffusion problems, due to the many attractive properties they exhibit. In particular, DGFEMs admit good stability properties, they offer flexibility in the mesh design (irregular meshes are admissible) and in the imposition of boundary conditions (Dirichlet boundary conditions are weakly imposed), and they are increasingly popular in the context of hp-adaptive algorithms. The increase in popularity for DGFEMs has created a corresponding demand for developing corresponding linear solvers. This work aims to provide an overview of the current state of affairs in the solution of DGFEM-linear problems and present some recent results on the preconditioning of stiffness matrices arising from DGFEM discretisations of steady-state convection-diffusion boundary-value problems. More specifically, preconditioners are derived for which theoretical results indicate GMRES performance independent of discretisation parameters.

discontinuous Galerkin
finite element methods
preconditioning
Krylov methods
GMRES

http://resolver.tudelft.nl/uuid:c296334f-5a9d-4216-b208-e057c3199d16

Conference proceedings

conference paper

(c) 2006 Georgoulis, E.H.; Loghin, D.