A multilayer method of fundamental solutions for Stokes flow problems

The method of fundamental solutions (MFS) is a meshless method for the solution of boundary value problems and has recently been proposed as a simple and efficient method for the solution of Stokes flow problems. The MFS approximates the solution by an expansion of fundamental solutions whose singularities are located outside the flow domain. Typically, the source points (i.e. the singularities of the fundamental solutions) are confined to a smooth source layer embracing the flow domain. This monolayer implementation of the MFS (monolayer MFS) depends strongly on the location of the user-defined source points: On the one hand, increasing the distance of the source points from the boundary tends to increase the convergence rate. On the other hand, this may limit the achievable accuracy. This often results in an unfavorable compromise between the convergence rate and the achievable accuracy of the MFS. The idea behind the present work is that a multilayer implementation of the MFS (multilayer MFS) can improve the robustness of the MFS by efficiently resolving different scales of the solution by source layers at different distances from the boundary. We propose a block greedy-QR algorithm (BGQRa) which exploits this property in a multilevel fashion. The proposed multilayer MFS is much more robust than the monolayer MFS and can compute Stokes flows on general two- and three-dimensional domains. It converges rapidly and yields high levels of accuracy by combining the properties of distant and close source points. The block algorithm alleviates the overhead of multiple source layers and allows the multilayer MFS to outperform the monolayer MFS.

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