An unfitted RBF-FD method in a least-squares setting for elliptic PDEs on complex geometries

TL;DR

This paper introduces a modified least-squares RBF-FD method that simplifies node placement over complex geometries by allowing interpolation points to be placed in a bounding box, improving robustness and accuracy.

Contribution

The paper presents a novel unfitted RBF-FD approach that decouples node placement from domain shape, enhancing ease of use and performance on complex geometries.

Findings

Robustness of the method on complex 2D geometries.

Improved approximation error and runtime efficiency.

Successful extension to 3D geometries with convergence results.

Abstract

Radial basis function generated finite difference (RBF-FD) methods for PDEs require a set of interpolation points which conform to the computational domain $\Omega$. One of the requirements leading to approximation robustness is to place the interpolation points with a locally uniform distance around the boundary of $\Omega$. However generating interpolation points with such properties is a cumbersome problem. Instead, the interpolation points can be extended over the boundary and as such completely decoupled from the shape of $\Omega$. In this paper we present a modification to the least-squares RBF-FD method which allows the interpolation points to be placed in a box that encapsulates $\Omega$. This way, the node placement over a complex domain in 2D and 3D is greatly simplified. Numerical experiments on solving an elliptic model PDE over complex 2D geometries show that our approach is robust. Furthermore it performs better in terms of the approximation error and the runtime vs. error compared with the classic RBF-FD methods. It is also possible to use our approach in 3D, which we indicate by providing convergence results of a solution over a thoracic diaphragm.