The method of fundamental solutions for the Stokes flow with the subdomain technique
Rek, Zlatko ; Šarler, Božidar
The collocation version of the Method of Fundamental Solutions (MFS) with subdomains is introduced in the present work for the solution of the 2D Stokes flow in backward-facing-step geometry, ...including Dirichlet and Neumann boundary conditions. The motivation for the present work is the inability of the MFS to solve such problems and the problems with slits and cracks due to the discretization of a single domain. The inability stems from the artificial boundary that is difficult or impossible to properly geometrically set in such cases. The solution for such problems is found by splitting such domains into subdomains. The MFS equations for the equilibrium conditions at the collocation points on the interface between the adjacent subdomains are derived for the Stokes equation. A matrix that simultaneously solves the collocation problem on all the subdomains is formed and solved. A sensitivity study of the MFS results is performed by comparing the relative root mean square error with the reference solution obtained by the classical mesh-based finite volume method on a very fine mesh. The subdomain technique is verified by dividing the domain into 2, 3 and 5 subdomains. The velocity, vorticity and pressure compare very well with the reference solution in all three cases while the solution for the single domain approach is outstandingly poor and inappropriate. The paper shows that the proposed subdomain technique maintains the simplicity, true meshless character and accuracy of the MFS for the Stokes flow in cases where the domain topology requires the use of the subdomain technique.
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