Perturbation and stability analysis of strong form collocation with reproducing kernel approximation

Solving partial differential equations using strong form collocation with nonlocal approximation functions such as orthogonal polynomials and radial basis functions offers an exponential convergence, but with the cost of a dense and ill-conditioned linear system. In this work, the local approximation functions based on reproducing kernel approximation are introduced for strong form collocation method, called the reproducing kernel collocation method (RKCM). We perform the perturbation and stability analysis of RKCM, and estimate the condition numbers of the discrete equation. Our stability analyses, validated with numerical tests, show that this approach yields a well-conditioned and stable linear system similar to that in the finite element method. We also introduce an effective condition number where the properties of both matrix and right-hand side vector of a linear system are taken into consideration in the measure of conditioning. We first derive the effective condition number of the linear systems resulting from RKCM, and show that using the effective condition number offers a tighter estimation of stability of a linear system. The mathematical analysis also suggests that the effective condition number of RKPM does not grow with model refinement. The numerical results are also presented to validate the mathematical analysis. Copyright (C) 2011 John Wiley & Sons, Ltd.