High Order DGFEM for Elliptic Problems in 2d Polygons

Description

The Discontinuous Galerkin Finite Element Method (DGFEM) for stationary diffusion and linear elasticity problems with Dirichlet and Neumann boundary conditions is explored. The corresponding domains are assumed to be bounded polygons in two space dimensions. As it is well-known from the regularity theory of elliptic partial differential equations, the solutions of such problems may exhibit some singularities in the corners of the polygons and in the points of changing boundary condition type. In order to resolve these singularities numerically, the finite element meshes and the polynomial degree distributions for the DGFEM have to be chosen appropriately.

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The theoretical part of this project deals with the convergence properties of the h and the hp version of the DGFEM: optimal convergence of the h DGFEM and even exponential convergence of the hp DGFEM have been proved recently. Moreover, for diffusion problems, these theoretical results are confirmed with some numerical experiments. The corresponding results for linear elasticity problems are outstanding.

Interestingly, the DGFEM seems to achieve a higher level of accuracy than standard finite element methods (FEMs). Moreover, it is a great advantage that irregular meshes may be used for the DGFEM.

Future work:

• Implementation of the DGFEM for elasticity problems in MATLAB or C++
• DGFEM for the Stokes problem, etc.

Contacts

Prof. Christoph Schwab and Dr. Thomas P. Wihler.