A superconvergence result in the RBF-FD method

TL;DR

The paper investigates the error behavior of the RBF-FD method using Polyharmonic Splines augmented with monomials for solving the Poisson problem on the unit disk. By combining operator-truncation error analysis with Bayona's explicit interpolation-error formula, it shows that the local operator error scales as $h^{m-1}$ while the global system inversion yields a higher convergence order, specifically $h^{m}$, for even augmentation degrees $m$. This identifies a superconvergence phenomenon in a meshless, stencil-based discretization and explains it through term-by-term analysis of the error contributions after solving the global system. The findings inform the choice of augmentation degree in practice and provide a framework for understanding superconvergence in RBF-FD schemes for PDEs.

Abstract

Radial Basis Function-generated Finite Differences (RBF-FD) is a meshless method that can be used to numerically solve partial differential equations. The solution procedure consists of two steps. First, the differential operator is discretised on given scattered nodes and afterwards, a global sparse matrix is assembled and inverted to obtain an approximate solution. Focusing on Polyharmonic Splines as our Radial Basis Functions (RBFs) of choice, appropriately augmented with monomials, it is well known that the truncation error of the differential operator approximation is determined by the degree of monomial augmentation. Naively, one might think that the solution error will have the same order of convergence. We present a superconvergence result that shows otherwise - for some augmentation degrees, order of convergence is higher than expected.

Figures (4)

• Figure 1: Analytical solution $u(x,y)$, shown on an example discretisation generated during our solution procedure.
• Figure 2: Convergence of mean errors under decreasing $h$. Operator and solution errors are shown on the left and right, respectively for different monomial augmentation degrees $m$. For each $m$, the errors have been computed for 30 different discretisation sets. The results lie in the corresponding shaded regions.
• Figure 3: Solution and operator error dependences on scaling factor $R$ shown for different monomial augmentation degrees $m$. For each $m$, the errors have been computed for 30 different discretisation sets. The results lie in the corresponding shaded regions.
• Figure 4: Average error for each term in Bayona's formula for the operator approximation on the left, and the same terms after solving the global system on the right. Once again, the results over 30 different discretisations are shown. In both cases $m=2$ was fixed and first few leading terms in the formula were considered.