- FWF-Projekt P23484-N18: hp-FEM für Optimalsteuerungsprobleme
- Leader: Dr. Daniel Wachsmuth
- Co-Leader: Prof. Dr. Sven Beuchler
- duration: 36 months (starting soon)
- Projects members: Katharina Hofer
- Summary: The modeling of technical processes often leads to a description by partial differential equations.
Here, it becomes important to optimize these processes and its parameters, which results in the formulation
of an infinite-dimensional optimization problem. This optimization problem is often complemented by inequality constraints that
mimic technical limitations like for instance maximal temperatures.

Such optimization problems cannot be solved by hand in general. Hence, it is important to study finite-dimensional approximations that can be solved with the help of computers. Here it is crucial to employ efficient discretization schemes. In order to develop efficient discretizations it is essential to investigate and exploit the structure of the optimization problem. The solutions of inequality constrained optimization problems can be characterized by means of active and inactive sets, i.e. the information where inequality constraints are active (inequalities are satisfied with equality) or inactive (inequalities are strictly fulfilled). In particular, in the case of pointwise inequality constraints, the boundaries of active/inactive sets contain the singular part of the solution, while the solution is smooth otherwise.

The project will explore these structural information to develop efficient discretization methods. The main focus is to apply higher-order finite element methods that make use of information about active sets. A mix of a-priori and a-posteriori methods will be characteristic for the project: developing a-priori error estimates with respect to the number of unknowns and implementing a-posteriori discretization techniques.

- FWF Project P20121-N18: Schnelle hp-Löser für gemischte und elliptische Probleme
- Leader: Dr. Sven Beuchler
- Project member: Dipl.-Math techn. Martin Purrucker
- duration: 36 months (starting October 2007)
- Summary:
Many applications from science and engineering are mathematically described by partial differential equations.
The finite element method (FEM) is certainly the most powerful tool for the computer simulation of such models.
The p-version of the FEM operates on a fixed mesh, and increases the polynomial degree p per element.
The advantage of this method is that smooth functions can be approximated very well by high order polynomials.
Thus, the p-, and the hp- version of the FEM have become very popular discretization methods in mathematics and engineering.

The discretization of a boundary value problem using FEM leads to a linear system of algebraic equations Ax=b. It is known from the literature that preconditioned Krylov subspace methods are among of the most efficient iterative solution methods for Ax=b. The convergence speed of this methods depends strongly on the choice of the considered preconditioners.

In this project, several three dimensional boundary value problems will be discretized by the p- or the hp-version of the FEM using hexahedral elements. The corresponding linear system Ax=b will be solved by a preconditioned Krylov subspace method. We will develop several domain decomposition preconditioners for the system Ax=b such that the total solver time of Ax=b is proportionally to the dimension of the matrix A, i.e. the number of unknowns. We intend to use the tensor product structure of the hexahedral elements for the development of the preconditioners.

We will investigate preconditioners for hp-FEM discretizations of scalar elliptic problems as well as for hp-FEM discretizations of the Lame equations of linear elasticity. Moreover, we will consider the Stokes problem as an example of a mixed problem. All preconditioners will be investigated theoretically and numerically in several numerical experiments.