Adaptive meshless refinement schemes for RBF-PUM collocation

Abstract

In this paper we present an adaptive discretization technique for solving elliptic partial differential equations via a collocation radial basis function partition of unity method. In particular, we propose a new adaptive scheme based on the construction of an error indicator and a refinement algorithm, which used together turn out to be ad-hoc strategies within this framework. The performance of the adaptive meshless refinement scheme is assessed by numerical tests.

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

“…Indeed, one of the advantages of the PU scheme is that it might oversample only regions where the solution has high variation, e.g. in our case the patch where the singularity is located [50]. However, the oversampling might cause instability problems due to ill-conditioning of the local collocation matrices.…”

An Efficient Trivariate Algorithm for Tetrahedral Shepard Interpolation

“…Then, once the data points are stored in such blocks, an optimized searching technique is applied to detect the nearest neighbor points, thus enabling us to carry out a suitable choice of tetrahedra for 3D interpolation. Similar techniques were also studied in [7,4,5] in the context of partition of unity methods combined with the use of local radial basis functions, and suitably adapted to 2D interpolation via triangular Shepard interpolants [6]. Note that in this work we present the tetrahedral Shepard method and the related theoretical results for a generic domain Ω ⊂ R 3 .…”

“…Similar computational procedures were first introduced to efficiently solve interpolation and differential problems in the context of radial basis function partition of unity methods [7,4,5], and then suitably modified for classical and triangular Shepard-type methods defined on plane, sphere and other manifolds [1,6]. Although such fast algorithms can be applied to 2D and 3D generic domains, for the sake of brevity and clarity, here we describe our localizing and searching techniques focusing on the unit cube, that is the domain Ω = [0, 1] 3 ⊂ R 3 .…”

Adaptive Radial Basis Function Partition of Unity Interpolation: A Bivariate Algorithm for Unstructured Data

“…The original idea of PUM comes from the context of PDEs [3,27], but later it also gained much popularity in the field of numerical approximation [10,11,13] and, more in general, in various areas of applied mathematics and scientific computing (see e.g. [4,5,12,19,22,23,25,28]). However, although the RBF-PUM has some specific features that makes it particularly suitable for processing large scattered data sets, the problem of interpolating very irregularly distributed data points has been addressed only partly in [13].…”