Optimal Numerical Basis Functions in the Partition of Unity Method

Abstract

Partial differential equations (PDE) arise from the modeling of a wide range of physical problems and knowing how to solve them is of large interest in many industrial undertakings. One numerical method to solve PDE is the Partition of Unity Method (PUM) is based on a cover consisting of overlapping patches and independent local (patch-wise) approximation spaces. The global error of the PUM is a direct consequence of the local errors, which hence should be as small as possible. Instead of relying on heavy spatial refinement, the approximation quality of the PUM space can be enhanced by adding enrichment functions, which has the potential to substantially reduce the number of degrees of freedom required for an adequate discretization of a PDE under study. This thesis proposes details of a constructive method to compute optimal local approximation spaces, which can then be used as enrichment spaces in the PUM. The original framework, which had been introduced for the case of second-order elliptic PDE, was generalized to the case of even-order elliptic PDE. The optimal basis functions can be pre-computed in an offline phase, depend on the partial differential operator, but are independent of the explicit values of load and boundary conditions appearing in the problem. During the writing of this thesis, algebraic conditions ensuring geometric reusability of the optimal basis functions, whose computation is numerically expensive, were developed. In order to better understand the impact of various parameters on the performance of the optimal basis functions in a global enriched simulation, a series of benchmark problems with a yet computable reference solution to compare against were investigated. Finally, two additional proof-of-concept problems are solved with the help of optimal local approximation spaces. While these problems require millions of degrees of freedom in traditional, mesh-based methods, solutions with a fine level of detail can be computed with only a relatively small number of degrees of freedom in the PUM using optimal local approximation spaces. This shows, that optimal local basis functions can be used as a viable alternative for direct spatial refinement, and consequently permit the enriched PUM to provide detailed solutions to problems which were at best solvable with a very low accuracy using traditional methods.