A stabilized RBF collocation scheme for Neumann type boundary value problems

Abstract

The numerical solutionof partial dif- ferential equations (PDEs) with Neumann bound- ary conditions (BCs) resulted from strong form collocation scheme are typically much poorer in accuracy compared to those with pure Dirichlet BCs. In this paper, we show numerically that the reason of the reduced accuracy is that Neu- mann BC requires the approximation of the spa- tial derivatives at Neumann boundaries which are significantly less accurate than approximation of main function. Therefore, we utilize boundary treatment schemes that based upon increasing the accuracy of spatial derivatives at boundaries. In- creased accuracy of the spatial derivative approx- imation can be achieved by h-refinement reduc- ing the spacing between discretization points or byincreasingthemultiquadricshape parameter, c. Increasing the MQ shape parameter is very com- putationally cost effective, but leads to increased ill-conditioning. We have implemented an im- proved version of the truncated singular value de- composition (IT-SVD) originated by Volokh and Vilnay (2000) that projects very small singular values into the null space, producing a well con- ditioned system of equations. To assess the pro- posed refinement scheme, elliptic PDEs with dif- ferent boundary conditions are analyzed. Com- parisons that made with analytical solution reveal superior accuracy and computationalefficiency of the IT-SVD solutions.