Error Analysis of Nodal Meshless Methods

TL;DR

This paper develops a method to numerically compute explicit error bounds for various nodal meshless discretizations of elliptic boundary value problems, enabling comparison of different methods without requiring exact solutions.

Contribution

It introduces a way to explicitly calculate error bounds for any nodal meshless discretization, improving error analysis and comparison of methods.

Findings

Error bounds can be computed numerically for various discretizations.

The bounds are sharp under mild assumptions.

Numerical examples demonstrate the method's effectiveness.

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 $x_1,\ldots,x_M\in \Omega\subset\R^d$. 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 $$ \max_{1\leq j\leq M}|u^*(x_j)-\tilde u_j|\leq \epsilon \|u^*\|_S $$ where $\epsilon$ depends crucially on the problem and the discretization, but not on the solution. This contribution shows how to calculate such $\epsilon$ {\em 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.