# Fast tensor product solvers for hp-FEM using hexahedral elements

## Priv.-Doz. Dr. Sven Beuchler

April 24, 2007, 3:30 p.m. HF 136

In this talk, we investigate the hp-FEM discretization of an elliptic boundary value problem in 3D. The corresponding linear system is solved by a preconditioned conjugate gradient method with domain decomposition (DD) preconditioners. The key ingredient of the preconditioner is a polynomial basis which is stable in $L_2(-1,1)$ and $H^1(-1,1)$. In [1], a stable polynomial basis in $L_2$ and $H^1$ has been developed for the interior bubbles in $[-1,1]$. In this presentation, we generalize this result to the case of all polynomials. Using the tensor product structure of hexahedral elements, an efficient DD-preconditioner can easily be derived. Some numerical experiments show the efficiency of the proposed method.

This is a joint work with J. Schöberl (Aachen).