Error Analysis of Nodal Meshless Methods

Abstract

There are many application papers that solve elliptic boundary value problems by meshless methods, and they use various forms of generalized stiffness matrices that approximate derivatives of functions from values at scattered nodes $x_{1},\ldots,x_{M} \in \Omega \subset \mathbb{R}^{d}$. If u ^∗ is the true solution in some Sobolev space S allowing enough smoothness for the problem in question, and if the calculated approximate values at the nodes are denoted by $\tilde{u}_{1},\ldots,\tilde{u}_{M}$, the canonical form of error bounds is

$$\displaystyle{\max _{1\leq j\leq M}\vert u^{{\ast}}(x_{ j}) -\tilde{ u}_{j}\vert \leq \epsilon \| u^{{\ast}}\|_{ S}}$$

where ε depends crucially on the problem and the discretization, but not on the solution. This contribution shows how to calculate such ε numerically and explicitly, for any sort of discretization of strong problems via nodal values, may the discretization use Moving Least Squares, unsymmetric or symmetric RBF collocation, or localized RBF or polynomial stencils. This allows users to compare different discretizations with respect to error bounds of the above form, without knowing exact solutions, and admitting all possible ways to set up generalized stiffness matrices. The error analysis is proven to be sharp under mild additional assumptions. As a byproduct, it allows to construct worst cases that push discretizations to their limits. All of this is illustrated by numerical examples.