An Extended Error Analysis for a Meshfree Discretizatiion Method for Darcy's Problem

Abstract

Recently [D. Schräder and H. Wendland, Math. Comp., 80 (2011), pp. 263–277], a new meshfree approximation method for Darcy’s problem has been introduced and analyzed. This method is based on a symmetric collocation approach using radial basis functions producing solutions with an analytically divergence-free velocity part. However, the error analysis provided in that paper works only for smooth solutions, where the smoothness is intrinsically linked to the smoothness of the employed basis function. In this paper, we will extend the error analysis to less smooth functions, showing that the approximation order for rougher solutions is determined by the smoothness of the solution rather than the smoothness of the basis function.