Method of fundamental solutions with weighted average condition and dummy points

Abstract

The aim of this paper is to develop the method of fundamental solutions using weighted average condition and dummy points. We accomplish mathematical analysis, a unique existence of an approximate solution and an exponential decay of the approximation error, for a potential problem in disk, and show some numerical experiments, which exemplify our error estimate.

Structure‐preserving numerical scheme for the one‐phase Hele‐Shaw problems by the method of fundamental solutions

“…A feature of our scheme is to realize CS, AP, and BF properties in a discrete sense using the normal velocity determined via a modified invariant scheme of the MFS, so‐called Murota's invariant scheme, and the tangential velocity determined by the MFS and the uniform distribution method (UDM) . Notably, under the UDM, we have stable numerical computation.…”

“…To the best of our knowledge, application of the MFS for moving boundary problems is quite a few, for instance, the MFS for a stationary-free boundary problem for an irrotational flow of a perfect fluid 19 and the MFS combined with the level set method for the exterior Hele-Shaw problem without surface tension. 20 A feature of our scheme is to realize CS, AP, and BF properties in a discrete sense using the normal velocity determined via a modified invariant scheme of the MFS, so-called Murota's invariant scheme, [21][22][23][24] and the tangential velocity determined by the MFS and the uniform distribution method (UDM). 25 Notably, under the UDM, we have stable numerical computation.…”

On invariance of schemes in the method of fundamental solutions

“…In [9,10,11], the authors modified MFS for the potential problem in the plane by adding the dummy points {z k } N k=1 ⊂ R 2 \ Ω, and considering an approximate solution of the form…”

“…Therefore, we can add one more condition to determine coefficients such as weighted average condition [10]:…”