Adaptive Time-Space-discretizations in 4 Applications

Non-linear time-depending problems are still challenges in scientific computing even on supercomputers. Their effective numerical treatment requires a smart discretization in time and space and the two mechanisms interact. In particular, singularities or free boundaries in phase transitions, contact problems, or for American options require a highly refined mesh in some, a priori unclear, regions of the time-space domain. The detection of those areas and the proper treatment of the overall discretization is the topic of this project. Four non-standard model equations are in the simplest situation reasonable enough to capture characteristic phenomena which represent four innovative fields of applications in satellite dynamics, phase field evolution, quantum mechanics, and computational finance. All mathematical models are treated with similar general mathematical arguments and empirically studied in numerical experiments. The results of this project will enable more efficient algorithms for more complicated and innovative simulations in the future.

The understanding of adaptive methods for physical and engineering applications is of great importance. For the development of adaptive spatial discretisations a priori and a posteriori error estimates are important for the efficiency and exactness of the numerical simulations. As non-adaptive methods are rather costly, it usually pays to do some new adaptive spatial discretisation methods. The project "Adaptive spatial discretisations in 4 examples" funded by the FWF under grant P16461-N12, project 3230 7401, was concerned with the mathematical foundation and development of adaptive methods for spatial discretisations and their applications for real-life problems. A spatial discretisation method is adaptive if it the underlying domains are refined or coarsed with respect to their discretisation error e h = u - uh . The numerical simulation provides a discrete approximation uh of u, which depends on the refinement of the given triangulation. These algorithms allow the computation of an approximation uh of u up to given tolerance. Therefore, the a priori error estimates give an upper bound of the error e h depending from the exact solution u, where the a posteriori error-estimates gives an upper bound of the error e h depending only on computable quantities, for example e h . The 4 examples are related to the nonlinear partial differential equation, e.g., the evaluation of American Options, mathematical model of a tethered satellite systems (TSS), models for phase field and quantum mechanics. The problems are solved with time-space discretisations. One numerical challenging point in the discretisation is the necessary finding of duality and energy arguments. Therefore, standard Sobolev spaces have to extended to weighted Sobolev spaces for the examples in American options and also for the other models based on time-space discretisations. With these extensions it was possible to develop error-estimates and improved approximations methods. Obviously, this provides and allows more efficient and larger simulations. The numerical evidence raised within the project clearly predicts that an extension for the standard Sobolev-spaces allows new adaptive spatial discretisation methods. These methods will play an important tool in the future.