Error Analysis of Nodal Meshless Methods
For researchers and practitioners using meshless methods, this provides a practical tool to evaluate and compare discretization errors, though it is an incremental improvement over existing error analysis frameworks.
This paper presents a method to numerically and explicitly compute error bounds for nodal meshless methods solving elliptic boundary value problems, enabling comparison of different discretizations without knowing exact solutions. The analysis is proven sharp and illustrated with numerical examples.
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 $x_1,\ldots,x_M\in Ω\subset\R^d$. 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 $$ \max_{1\leq j\leq M}|u^*(x_j)-\tilde u_j|\leq ε\|u^*\|_S $$ where $ε$ depends crucially on the problem and the discretization, but not on the solution. This contribution shows how to calculate such $ε$ {\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.