RBF-Based Partition of Unity Methods for Elliptic PDEs: Adaptivity and Stability Issues Via Variably Scaled Kernels

Abstract

We investigate adaptivity issues for the approximation of Poisson equations via radial basis function-based partition of unity collocation. The adaptive residual subsampling approach is performed with quasi-uniform node sequences leading to a flexible tool which however might suffer from numerical instability due to ill-conditioning of the collocation matrices. We thus develop a hybrid method which makes use of the so-called variably scaled kernels. The proposed algorithm numerically ensures the convergence of the adaptive procedure.

A stable meshfree PDE solver for source-type flows in porous media

“…In the experiments, we consider nested sets of Halton data, with N = (13 + k) 2 , k = 0, 10, 20, 30. Of course, since the tolerance used to test if the local matrices are singular increases when the shape parameter decreases, as in [25], we expect that the accuracy of the HVSK-PU scheme significantly improves for small values of the shape parameters. On the opposite, the solution in terms of rRMSE should be comparable with that furnished by the classical PU scheme for ε = 2.…”

“…, N j . -the tolerance in (7), which as in [25] is linked to the shape parameter ε, as 10 −16 /ε 4 . In order to not further burden the computational cost, we fix h j i = 7 × 10 −6 , i = 1, .…”

“…To partially overcome this, we introduce the PU method. Taking advantage of the local scheme we can introduce a hybrid method that enables us to construct an accurate and stable numerical tool [25].…”

“…Here we investigate on the performances, in terms of stability and accuracy, of some collocation schemes on non-uniform distributions and we deal with the stability issues due to the ill-conditioning of the collocation matrices. To address this, we construct a hybrid method [25,26] with the use of Variably Scaled Kernels (VSKs), introduced in [27] and further investigated in [28,29]. Since VSKs work with any kernel, the so-constructed scheme turns out to be flexible, stable and accurate.…”

Solving 3-D Gray–Scott Systems with Variable Diffusion Coefficients on Surfaces by Closest Point Method with RBF-FD

“…The study of this parameter is a crucial topic to guarantee the stability of numerical methods. To overcome the problem, hybrid approaches called variably scaled kernels (VSKs) have been recently introduced [22].…”