Convergence theories and adaptive algorithms for Kansa Methods

To solve and investigate a real-life problem, a mathematical model is often constructed, which may involve differential equations and certain conditions. The traditional numerical solution procedure for such a model decomposes the region of interest into small sub- regions defined by a mesh, from relationships between these sub-regions. This research project in applied and computational mathematics is focused on whether a mathematical model can be solved without the need for meshing?

Although a meshed landscape, car or airplane may have aesthetic appeal for some, unfortunately the construction of a mesh is a “bottleneck” in many computations. Thus, although the actual computations may take few hours to complete, the mesh may take days or weeks to detail for complex geometries. So, there has been considerable recent interest in a new class of computational methods that do not require meshes – i.e., meshless methods. Since 2000, the PI for this project has been working intensively on developing the relevant algorithmic theory and on various applications of meshless methods. The aim of this research project is to develop a new algorithm that can avoid any mesh construction and exploit adaptive processes, for the convenience of the user who seeks a robust and efficient computational solution of a mathematical model.

An important missing aspect of the meshless asymmetric technique introduced by Kansa, designed for models involving time-independent linear partial differential equa- tions (PDE), is a solid mathematical basis for solution convergence. In this project, it is intended to provide the scientific and engineering communities with stable numerical al- gorithms for this method. The primary aim is to develop both theory and algorithms for the constrained least-squares Kansa method. The theoretical part is to determine how good a numerical solution could be, due to features such as the approximation power and convergence rate of these meshless approaches. On the practical side, proper im- plementations and efficient algorithms to realize the theoretically proven advantages will be provided. Further appropriate techniques such as adaptive data points refinement, adaptive greedy algorithms in the meshless numerical differentiation scheme will play an important role.