# Johannes Kepler Symposium on Mathematics

As part of the Johannes Kepler symposium on mathematics
**Priv.-Doz. Dr. Johannes
Kraus**, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences (ÖAW),
will give a public talk (followed by a discussion) on **Wed, March 11, 2009**
at **16:15 o'clock** at **HF 9901**
on the topic of
"Algebraic multilevel iteration methods for solving elliptic finite element equations"
. The organziers of the symposium,

Univ.-Prof. Dr. Gerhard Larcher

A.Univ.-Prof. Dr. Jürgen Maaß, and

die ÖMG (Österreichische Mathematische Gesellschaft),

hereby cordially invite you.

Series B - Mathematical Colloquium:

The intention is to present new mathematical results for an audience interested in general mathematics.

### Algebraic multilevel iteration methods for solving elliptic finite element equations

In this talk algebraic multilevel iteration (AMLI) methods for solving elliptic finite element equations with symmetric positive definite (SPD) matrices will be discussed. We start with the formulation of the linear AMLI algorithm and recall the classical convergence theory. In addition, we will address nonlinear multilevel preconditioners. We will also discuss complexity issues and comment on the recent developments in extending the theory of AMLI methods to nonconforming finite element discretizations.

The second part of the talk will be on multilevel methods for discontinuous Galerkin (DG) discretizations and variational problems in H(div). In particular, we show how the efficient AMLI methods for solving H(div) problems lead to an efficient computation of the functional-type a posteriori error estimates for DG problems.

Finally, we will present an example for integrating AMLI methods in the solution of more complex linear elasticity problems for nearly incompressible materials. In fact, the optimal solution of such nearly singular problems is also an essential part in the design of optimal solvers for indefinite systems. The robust AMLI algorithms that we introduce provide the key component of a solver for the indefinite systems obtained via the augmented Lagrangian method, and thus give a tool for solving the discretized Stokes equations and systems resulting from mixed finite element discretizations.