On improving the conditioning of the method of fundamental solutions for biharmonic BVPs in 2D domains

The MFS-SVD approach introduced in [1], which combines the method of fundamental solutions (MFS) with singular value decomposition (SVD), is a potential alternative to existing methods for improving the conditioning of MFS linear systems. However, until now, the feasibility of this approach for boundary value problems (BVPs) defined on planar domains, has only been illustrated for two problems involving second order partial differential equations: The Laplace equation in [1] and the homogeneous Helmholtz equation in [2]. These papers suggest that SVD should be applied to the ill-conditioned factor of the MFS linear systems decomposition, which does not apply to higher order problems. In this work, we bring more clarity to this point, contributing to the establishment of a complete procedure to follow when solving problems using the MFS-SVD approach. We use the biharmonic boundary value problem, a fourth order PDE, to illustrate this procedure. This is done by approaching the numerical solution of the problem using two different ansatz, which means two different addition theorems and two different decompositions, with the aim of reinforcing the idea about the robustness of the MFS-SVD, regardless of the numerical formulation being considered. As expected, the MFS-SVD performs similarly in both cases.