Numerical examination of the effect of different boundary conditions on the method of approximate particular solutions for scalar and vector problems

This work presents a numerical study of the influence of different boundary conditions (BC) on the precision of the method of approximate particular solutions (MAPS) with radial basis functions in two dimensions. The state of the art merely mentions that the MAPS accuracy decreases near the boundaries. Therefore, the influence that the choice of boundary condition has on this method should be studied with more detail. The present analysis with numerical experiments was applied to the solution of homogeneous Stokes flow equations, backward facing step flow, Slip flow, convective flow between parallel vertical plates and to elliptic differential equations with variable coefficients. The results show that care must be taken in choosing the boundary conditions for this type of solution methods because this selection affects the precision of the results. It is shown that the error associated with Neumann boundary condition, can be reduced by using a refined nodal distribution towards the inlet and outlet zones. Conversely, mixed boundary conditions require careful combination of traction and velocity to achieve better accuracy with MAPS for some flow problems.