The method of fundamental solutions for the numerical solution of the biharmonic equation
Journal of Computational Physics
Asymptotic stability of the fundamental solution method
ISCM '90 Proceedings of the International Symposium on Computation mathematics
Some Aspects of the Method of Fundamental Solutions for Certain Harmonic Problems
Journal of Scientific Computing
Novel meshless method for solving the potential problems with arbitrary domain
Journal of Computational Physics
Short Note: The method of fundamental solutions for 2D and 3D Stokes problems
Journal of Computational Physics
Journal of Computational and Applied Mathematics
Journal of Computational Physics
An adaptive greedy technique for inverse boundary determination problem
Journal of Computational Physics
Mathematics and Computers in Simulation
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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.