Oscillatory behaviour of the RBF-FD approximation accuracy under increasing stencil size

TL;DR

Focusing on Polyharmonic Spline RBFs with monomial augmentation, it is found that the solution error oscillates under increasing stencil size, and can connect this behaviour with the spatial dependence of the signed approximation error.

Abstract

When solving partial differential equations on scattered nodes using the Radial Basis Function generated Finite Difference (RBF-FD) method, one of the parameters that must be chosen is the stencil size. Focusing on Polyharmonic Spline RBFs with monomial augmentation, we observe that it affects the approximation accuracy in a particularly interesting way - the solution error oscillates under increasing stencil size. We find that we can connect this behaviour with the spatial dependence of the signed approximation error. Based on this observation we are then able to introduce a numerical quantity that indicates whether a given stencil size is locally optimal.

Figures (6)

• Figure 1: The analytical solution to the considered Poisson problem.
• Figure 2: Dependence of the approximation errors on the stencil size $n$.
• Figure 3: Behaviour of the approximation errors under a refinement of the discretisation.
• Figure 4: The seperation of the domain into two regions is seen on the left, where the green circles show the radii of the biggest stencils considered ($n=69$). The right graph shows the error dependence when either of the regions is at a fixed stencil size $n=28$. The previous result with no fixed stencil size regions is also shown.
• Figure 5: Spatial dependence of $e_\mathrm{poiss}^\pm$ in some local extrema. The colour scale is the same for all drawn plots.