# RBF Multiscale Collocation for Second Order Elliptic Boundary Value Problems

@article{Farrell2013RBFMC, title={RBF Multiscale Collocation for Second Order Elliptic Boundary Value Problems}, author={Patricio Farrell and Holger Wendland}, journal={SIAM J. Numer. Anal.}, year={2013}, volume={51}, pages={2403-2425} }

In this paper, we discuss multiscale radial basis function collocation methods for solving elliptic partial differential equations on bounded domains. The approximate solution is constructed in a multilevel fashion, each level using compactly supported radial basis functions of smaller scale on an increasingly fine mesh. On each level, standard symmetric collocation is employed. A convergence theory is given, which builds on recent theoretical advances for multiscale approximation using…

## 30 Citations

### A Multiscale RBF Collocation Method for the Numerical Solution of Partial Differential Equations

- Mathematics, Computer ScienceMathematics
- 2019

The hierarchical radial basis functions method for the approximation to Sobolev functions and the collocation to well-posed linear partial differential equations can not only solve the present problem on a single level with higher accuracy and lower computational cost, but also produce a highly sparse discrete algebraic system.

### Solving partial differential equations with multiscale radial basis functions

- Mathematics, Computer Science
- 2018

In this paper, both collocation and Galerkin approximation are described and analysed, and error estimates for both schemes are derived, though special emphasis is given to Galerkins approximation, since the current situation here is not as clear as in the case of collocation.

### Block preconditioners for linear systems arising from multilevel RBF collocation

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- 2014

It is shown that certain block diagonal and triangular preconditioners based on a restricted additive Schwarz method with coarse grid correction (ARASM) with high computational cost and high spectra deteriorate with the number of data points, particularly the condition number and sparsity.

### Multilevel sparse grids collocation for linear partial differential equations, with tensor product smooth basis functions

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### On Multiscale RBF Collocation Methods for Solving the Monge–Ampère Equation

- Computer Science
- 2020

The hierarchical radial basis function method can not only solve the present problem on a single level with higher accuracy and lower computational cost but also produce highly sparse nonlinear discrete system.

### Multilevel RBF collocation method for the fourth-order thin plate problem

- EngineeringInt. J. Wavelets Multiresolution Inf. Process.
- 2021

This paper uses nonsymmetric Kansa multilevel radial basis function collocation method to solve the fourth-order thin plate problem and examines that the convergence of the multileVEL radial basis functions meshfree collocations method which is good for solving theFourth-orderthin plate problem.

### Block preconditioners for linear systems arising from multiscale collocation with compactly supported RBFs

- Computer ScienceNumer. Linear Algebra Appl.
- 2015

It is shown that certain block diagonal and triangular preconditioners, based on a restricted additive Schwarz method with coarse grid correction, can be used for symmetric collocation methods with RBFs without needing to generate a grid.

### Adaptive meshless refinement schemes for RBF-PUM collocation

- Computer ScienceAppl. Math. Lett.
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### A Petrov-Galerkin Kernel Approximation on the Sphere

- Mathematics, Computer ScienceSIAM J. Numer. Anal.
- 2018

In this paper, a numerical solution of partial differential equations on the unit sphere is given by using a kernel trial approximation in combination with a special Petrov--Galerkin test…

### Multilevel sparse grid kernels collocation with radial basis functions for elliptic and parabolic problems

- Computer Science
- 2017

This thesis modifications the multilevel sparse grid kernel interpolation (MuSIK) algorithm proposed in [48] for use in Kansa’s collocation method (referred to as MuSIK-C) to solve elliptic and parabolic problems and shows similar performance in low dimension situation and better approximation in high dimension.

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