Adaptive Finite Elements for Parabolic Partial Differential Equations

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During the last years, computer science and scientific computing has become an important research branch located between applied mathematics, applied sciences, and engineering. Nowadays, in numerical mathematics not only simple model problems are treated, but sophisticated and well-founded mathematical algorithms are applied to solve complex problems of real life applications. Usually, the design of such algorithms is based upon analytical properties and a precise numerical analysis of the underlying problem. Real life applications are demanding for computational realization and need suitable and robust tools for a flexible and efficient implementation.Inspired by and parallel to the investigation of real life applications, numerical mathematics has built and improved many modern algorithms which are now standard tools in scientific computing. Examples are adaptive methods, higher order discretizations, fast linear and non-linear iterative solvers, multi-level algorithms, etc. These mathematical tools are able to reduce computing times tremendously and for many applications a simulation can only be realized in a reasonable time-frame using such highly efficient algorithms. This holds especially true for time-dependent parabolic problems that are the main focus of this research project. The aim of the research is the design and analysis of efficient numerical methods for this class of problems. We want to perform basic research on adaptive finite element methods. Nowadays, adaptive methods for stationary problems are well understood and well analyzed. This includes convergence and optimality of practically used algorithms. Adaptive methods for time-dependent problems are still based upon heuristics. Convergence and optimality of such algorithms is still an open problem. Team Siebert mainly focuses on convergence aspects of finite elements which includes improvement of existing estimators and design of new adaptive methods. Team Hoppe analyses control and state constrained optimal control problems for parabolic PDEs and team Funken focuses on time-periodic micromagnetic problems. With the envisaged research project we want to combine the expertise of the three working groups located at the Universities of Augsburg and Ulm. There will be a close cooperation of all teams with respect to a posteriori error analysis and suitable adaptive methods. To our best knowledge, this will be world-wide the only bundled research project with sole focus on adaptive methods for parabolic problems. We expect that the planned interaction of analysis, different discretization methods, and application to various problems leads to new robust and efficient numerical methods also for real life problems.