High-order FEM for optimal control problems

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 high-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.

The project investigated the efficient solution of optimal control problems subject to partial differential equations. Since partial differential equations cannot be solved explicitly, these equations have to be discretized, which transforms them into nonlinear equations. The resulting discrete optimal control problems are large-scale optimization problems with equality and inequality constraints. Taylored solution approaches can be devised exploiting the particular problem structure.The project concentrated on a special discretization technique: the so-called hp-finite element method. The principal idea is that the solution of the partial differential equation is approximated by piecewise polynomial functions with varying degree. During the projects lifetime we analyzed different flavours of this method. Let us highlight the most important results.First, we studied boundary and interface concentrated finite element methods. There, the solution of the optimal control problem is approximated by small elements with low polynomial degree near boundaries and interfaces. This is motivated by the fact that boundaries and interfaces introduce singularities dominating the regularity of the solution. We could prove convergence rates, which show that d-dimensional problems can be solved by the hp-finite element method with an accuracy comparable to a standard finite element approach applied to d ? 1 dimensional problems.Second, we applied two different adaptive refinement techniques. There, the discretization is driven by the solution of the discrete problem itself. The algorithms are implemented into an experimental software. Numerical tests show a refinement of the mesh along the interface as well as an expected convergence order for exact solutions which are smooth outside the interface.The most interesting result however is the proof of exponential convergence: We could show that for a special, a-priori chosen discretization, the discretization error decreases exponentially with respect to the number of unknowns. This is known to be optimal, as there cannot be any general-purpose technique that converges faster.Finally, we have developed optimal solvers for the systems of algebraic equations arising from the discretization of the optimal control problem in two and three space dimensions.