hp-Finite element approximations of discontinuous Galerkin type for flow problems

Description

Discontinuous Galerkin (DG) methods have a long history and have recently become more and more popular. They have been heavily tested and studied, and they present considerable advantages for certain types of problems, especially those modeling phenomena where convection is moderate or strong.

Their main idea relies in the choice of approximation spaces consisting of piece-wise polynomial functions with no kind of continuity constraints across the interface between the elements of a triangulation. Consistency and well-posedness are achieved by introducing suitable bilinear forms defined on the interface. In this respect they are closely related to finite volume methods as they relies on the definition of numerical fluxes. As for conforming finite element approximations, the corresponding discrete problem is given in terms of finite dimensional subspaces and bilinear forms.

One of the main advantages of DG methods is that they allow a much greater flexibility in the design of the mesh and in the choice of the approximation spaces. A mixed domain decomposition approach is also natural where conforming approximations are considered on single subdomains or patches, and DG interface terms are introduced on the boundaries between the subdomains.

While extensive work has been done for scalar diffusion or advection-diffusion problems in recent years, there are considerably fewer works for DG discretizations of less standard problems, such as saddle-point problems describing, e.g., nearly incompressible solids or incompressible fluid flows, electromagnetic problems, or scattering problems.

The purpose of this project is to develop efficient and accurate DG approximations for flow problems at various regimes. It consists of a theoretical part, where efficient methods are devised and analized, and a programming part, where the methods are implemented and tested on some real life problems.

Research Reports

• 2003-13 A numerical study on Neumann-Neumann methods for hp approximations on geometrically refined boundary layer meshes II: Three-dimensional problems
• 2003-09 Energy norm a-posteriori error estimation for mixed discontinuous Galerkin approximations of the Stokes problem
• 2003-01 Domain decomposition preconditioners of Neumann-Neumann type for hp-approximations on boundary layer meshes in three dimensions
• 2002-25 Mixed hp-DGFEM for incompressible flows III: Pressure stabilization
• 2002-21 Mixed hp-DGFEM for incompressible flows II: Geometric edge meshes
• 2002-10 Mixed hp-DGFEM for incompressible flow
• 2002-22 hp Finite element approximations on non-matching grids for the Stokes problem
• 2002-20 A numerical study on Neumann-Neumann and FETI methods for hp-approximations on geometrically refined boundary layer meshes in two dimensions
• 2002-15 Neumann-Neumann and FETI preconditioners for hp-approximations on geometrically refined boundary layer meshes in two dimensions
• 2002-02 hp discontinuous Galerkin approximation for the Stokes problem
• 2001-08 Convergence of some two-level overlapping domain decomposition preconditioners with smoothed aggregation coarse space
  2001-02 Mixed hp-finite element approximations on geometric edge and boundary layer meshes in three dimensions
• 2000-12 An overlapping domain decomposition preconditioner for a class of discontinuous Galerkin approximations of advection-diffusion problems

In collaboration with

• Prof. Dominik Schötzau, University of Basel

Contacts

Prof. Christoph Schwab and Dr. Andrea Toselli.