# Fast hp-solvers for quadrilateral and hexahedral elements

## Dipl.-Math. Martin Purrucker

**April 28, 2009, 1:30 p.m. MZ 005B**

The ﬁnite element method is a well known method for the numerical solution of elliptic partial diﬀerential equations. While the h-version concentrates on decreasing the diameter h of the elements in order to get more accurate results, the p-version deals with increasing the polynomial degree p of the basis functions on the elements. The combination of both is called hp-version.

In any case a large linear system has to be solved in a very eﬃcient way, i.e. applying the preconditioned conjugate gradient method with an eﬃcient preconditioner. This talk gives an overview of known results and ongoing research work on the construction of fast hp-solvers for quadrilateral (2D) and hexahedral elements (3D) based on domain decomposition (DD) methods.

In 2D a non-overlapping DD-preconditioner with inexact subproblem solvers is used. However, such a stable decomposition into the standard spaces is not possible in 3D. Instead a DD-preconditioner of Pavarino is considered, decomposing the space into a coarse space and patches related to each node, where on each patch the preconditioner of Korneev/Langer/Xanthis can be applied.