Approximation with Conditionally Positive Definite Kernels on Deficient Sets

Improved Stencil Selection for Meshless Finite Difference Methods in 3D

“…The above methods are applied only for selecting the sets of influence, while the weights w ζ,ξ of (3) are obtained by the RBF-FD method described in Section 3, with ϕ(r) = r 5 and ℓ = 3. On the other hand, pQR selection method introduced in [4] already generates weights satisfying the polynomial exactness condition (5), and corresponding meshless finite difference method performs well in the numerical experiments presented in [8]. Moreover, since the number of selected nodes in this case does not exceed the polynomial dimension L, RBF-FD weights computed with the polynomial term of order ℓ are only rarely different from the pQR weights of the same order.…”

“…Unfortunately, no selection method shows a good performance in this case. The methods 20near, oct and oct-dist fail to provide polynomial exactness (5). The errors of 30near, pQR4sel, pQR3 and pQR4 are shown in Table 11 and are disappointing.…”

“…, p L }, L = ℓ+2 3 , is a basis for the linear space of trivariate polynomials of order at most ℓ (that is, total degree at most ℓ − 1). As shown in [5], the RBF-FD weights w ζ,ξ are uniquely determined by these conditions as soon as Φ is positive definite or conditionally positive definite of order at most ℓ, and there is any solution…”

“…We refer to [5] for a discussion of computational methods for w ζ,ξ . As demonstrated in [1], particularly good results are obtained with polyhamonic RBF defined by ϕ(r) = r α with a polynomial term.…”

“…For this we employ MATLAB's command scatteredInterpolant with default settings, which makes use of piecewise linear interpolation over a triangulation of the nodes. Note that we write E ref = NaN in the tables below when the weights of Section 3 cannot be found for some ζ ∈ Ξ int , which only happens when there are no weights satisfying the polynomial exactness condition (5). On rare occasions we write E ref = Inf, which means that MATLAB failed to solve the system (3) because its matrix was singular to working precision.…”