Enrichment Techniques and the Method of Fundamental Solutions

Proceedings of Lsame.08: Leuven Symposium on Applied Mechanics in (Ed. Bergen, B et al.), (2008), 121-132
ISBN:978-90-73802-85-8

It is well known that the method of fundamental solutions (MFS) provides excellent numerical results while solving boundary value problems for linear elliptic differential equations on smooth boundaries and smooth data. However, despite the theoretical density results, when the solution is not smooth the performance of the MFS might be compromised by discontinuities on the boundary conditions or by the nonsmoothness of the boundary. Moreover, even with some boundary regular data, we may have poor approximations generated by classical choices for the artificial source points, for instance on a circle. In this work we discuss the long standing issue of collocating the source points in the MFS, and propose a method mixing local - global features of the MFS approach, that we will call glocal. This choice for the location of the source points is adapted not only to the geometry but also to the boundary problem data. This technique can be used together with standard enrichment, but it can also be seen as some type of enrichment since the justification for the sources position is based on the local approximation. This glocal approach to the MFS, together with basis enrichment (using particular solutions), keeps advantages of Trefftz methods and avoids larger systems, worse conditioned.