Project within the collaborate research center (SFB) 256 "Nonlinear partial differential equations".
In this project new numerical methods for the discretization and solution of partial differential equations will be developed, analyzed and backed up theoretically, as well as implemented an modern workstations and tested on practical problems. Special focus is hereby posed on meshless numerical methods for the treatment of three-dimensional incompressible flows with free boundaries, inner layers and multi-phase fluids. These have special properties like complicated geometries of the flow area varying with time, and effects and forces induced by the free boundary. These include for example numerical treatment of surface tension or Marangoni convection caused by temperature gradients (in the case of very thin fluid films or under micro gravitation).
Concretely these aspects will be examined and developed:
Meshless discretization methods using radial basis functions with compact support will be considered. No grids will be used for discretization, but single points which show no stiff connections. The matrices resulting from Galerkin approximation with radial basisfunctions as test functions will be examined and suitable data structures will be developed. Furthermore adaptive refinement methods for meshless discretizations are to be found. Finally the application to the Navier-Stokes equations with free boundaries using the approaches of Euler and Lagrange will be carried out.
The points sets arising from meshless discretizations can bee seen as approximations of continous densities. Using coarser approximations, e.g. by reduction of the number of used points (sparsify), and using radial basis functions with larger support coarser spaces can be generated. This leads to a multilevel approach where discrete, but in general not nested spaces aries and where no natural multilevel sequence is given. Nevertheless additive and multiplicative multilevel methods can be constructed. For this construction restriction and prolongation operators will be developed and robustness of the resulting multigrid/multilevel methods will be considered.
Methods for the adaptive representation and movement of interfaces and free boundaries for two- and multiphase flows in 3D will be developed, which are based on the meshless method. Additionally the idea of the hierarchical basis will be applied to the representaion of curves and surfaces. Furthermore, the evaluation of boundary conditions as well as terms and operators which live on the boundary, e.g. surface tension and Marangoni convection, will be realized.