# Multi-level solver for the p-Version of the FEM

## Priv.-Doz. Dr. Sven Beuchler

**March 20, 2003, 2:30 p.m. T 811**

From the literature it is known that the conjugate gradient method with do- main decomposition preconditioners is one of the most efficient methods for solving systems of linear algebraic equations resulting from p-version finite element discretizations of elliptic boundary value problems. The ingredients of such a preconditioner are a preconditioner for the Schur complement, a preconditioner related to the Dirichlet problems in the subdomains, and an extension operator from the boundaries of the subdomains into their interior. In the case of Poisson's equation, we propose a preconditioner for the problems in the subdomains which can be interpreted as the stiffness matrix resulting from an h-version finite element discretization of a degenerate operator.

- A multi-grid algorithm with a special line smoother,
- a wavelet preconditioner and
- an Algebraic Multi-Level Iteration preconditioner

are defined in order to solve the corresponding systems of finite element equations.