# Fast hp-solvers for quadrilateral and hexahedral elements with applications to the Poisson and Stokes problem

## Dipl.-Math. Martin Purrucker

**Nov. 24, 2009, 3:30 p.m. P 215**

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

In the first part of this talk an introduction to hp-FEM with quadrilateral or hexahedral elements for the discretization of the Poisson problem is presented and the fast numerical solution of the discrete equation based on the preconditioned conjugate gradient method with domain decomposition preconditioners is discussed.

In the second part of this talk the Stokes boundary value problem is discretized with hp-FEM with quadrilateral or hexahedral elements. Starting from a mixed variational formulation it is well known that the discrete FE-spaces for the velocity and the pressure have to be chosen in such a way that a discrete inf-sup-condition holds in order to get useful results. Finally the efficient numerical solution of the discretized mixed problem is investigated.