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# Non-Singular Method of Fundamental Solutions based on Laplace decomposition for 2D Stokes flow problems

E. Sincich1, B. Šarler1,2,3
1 University of Nova Gorica, Nova Gorica, Slovenia.
2 Institute of Metals and Technology, Ljubljana, Slovenia.
3 Corresponding author: Professor Božidar Šarler (bozidar.sarler@ung.si, bozidar.sarler@imt.si)

Computer Modeling in Engineering & Sciences 2014, 99(5), 393-415. https://doi.org/10.3970/cmes.2014.099.393

### Abstract

In this paper, a solution of a two-dimensional (2D) Stokes flow problem, subject to Dirichlet and fluid traction boundary conditions, is developed based on the Non-singular Method of Fundamental Solutions (NMFS). The Stokes equation is decomposed into three coupled Laplace equations for modified components of velocity, and pressure. The solution is based on the collocation of boundary conditions at the physical boundary by the fundamental solution of Laplace equation. The singularities are removed by smoothing over on disks around them. The derivatives on the boundary in the singular points are calculated through simple reference solutions. In NMFS no artificial boundary is needed as in the classical Method of Fundamental Solutions (MFS). Numerical examples include driven cavity flow on a square, analytically solvable solution on a circle and channel flow on a rectangle. The accuracy of the results is assessed by comparison with the MFS solution and analytical solutions. The main advantage of the approach is its simple, boundary only meshless character of the computations, and possibility of straightforward extension of the approach to three-dimensional (3D) problems, moving boundary problems and inverse problems.

### Keywords

Stokes problem, Laplacian decomposition, non-singular method of fundamental solutions, Laplace fundamental solution.

Sincich, E., Šarler, B. (2014). Non-Singular Method of Fundamental Solutions based on Laplace decomposition for 2D Stokes flow problems. CMES-Computer Modeling in Engineering & Sciences, 99(5), 393–415. This work is licensed under a Creative Commons Attribution 4.0 International License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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