Location of Repository

Krylov-Subspace Preconditioners for Discontinuous Galerkin Finite Element Methods

By Emmanuil H. Georgoulis and Daniel Loghin


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 th

Publisher: Dept. of Mathematics, University of Leicester
Year: 2006
OAI identifier: oai:lra.le.ac.uk:2381/4269

Suggested articles



  1. (2005). Stopping criteria for iterations in finite element methods., doi
  2. (1982). An interior penalty finite element method with discontinuous elements, doi
  3. (2004). Discontinuous Galerkin methods for first-order hyperbolic problems., doi
  4. (1976). A generalized conjugate gradient method for nonsymmetric systems of linear equations, in doi
  5. (1982). Iterative methods for large sparse non-symmetric systems of linear equations, doi
  6. (1990). On the theory of equivalent operators and application to the numerical solution of uniformly elliptic partial differential equations, doi
  7. (2006). A multilevel domain decomposition method for discontinuous Galerkin methods. in preparation,
  8. (2003). Discontinuous Galerkin methods on shape-regular and anisotropic meshes, D.Phil. Thesis,
  9. (2006). Krylov-subspace preconditioners for hp-version discontinuous Galerkin finite element methods, in preparation, doi
  10. (2005). Optimal error estimates for the hp–version interior penalty discontinuous Galerkin finite element method, doi
  11. (2000). Stabilised hp-finite element methods for firstorder hyperbolic problems, doi
  12. (2002). hp-finite element methods for advection-diffusion-reaction problems, doi
  13. (2003). Preconditioning methods for local discontinuous Galerkin discretisations, doi
  14. (1003). An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems, doi
  15. (2004). Analysis of block preconditioners for saddle-point problems, doi
  16. (2000). Review of a priori error estimation for discontinuous Galerkin methods, doi
  17. (1996). Iterative Methods for Sparse Linear Systems, doi
  18. (1998). p- and hp- finite element methods: Theory and applications in solid and fluid mechanics, doi

To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.