Workshop on Foundations of Numerical PDEs

July 7 - July 9, 2005

Universidad de Cantabria in Santander, Spain

Boundary Concentrated FEM

Jens Melenk

Mathematics at University of Reading

Abstract:   It is well-known for elliptic problems with smooth coeffcients that the solution is smooth in the interior of the domain; low regularity is only possible near the boundary. The hp-version of the finite element method (hp-FEM) with variable order polynomial degree distribution allows us to exploit this observation to get optimal (in the sense of n-widths) approximation methods by using meshes where the element size grows porportionally to the element?s distance to the boundary and the approximation order is suitably linked to the element size. In this way most degrees of freedom are concentrated near the boundary, and whence comes the name of this variant of the hp-FEM. A focus of this talk, will be (near) optimal solution techniques for the arising linear system. The first approach we discuss is based on multilevel techniques for this variant of the hp-FEM. The second approach considered is based on the concept of H-matrices, recently introduced by W. Hackbusch. The boundary concentrated FEM is variant of the hp-FEM; many of the implementation issues discussed for it apply in fact to the hp-FEM in general.