High-order numerical method for solving elliptic partial differential equations on unfitted node sets

TL;DR

This paper introduces a high-order numerical method using radial basis function-generated finite differences on unfitted node sets, enabling accurate solutions of elliptic PDEs in arbitrary domains without boundary-fitting.

Contribution

It proposes a novel RBF-FD based approach with boundary conditions enforced via Lagrange multipliers, offering high-order accuracy and geometric flexibility.

Findings

Achieves high-order accuracy in arbitrary domains

Demonstrates robustness through numerical experiments

Enables boundary conditions enforcement without boundary fitting

Abstract

In this paper, we present how high-order accurate solutions to elliptic partial differential equations can be achieved in arbitrary spatial domains using radial basis function-generated finite differences (RBF-FD) on unfitted node sets (i.e., not adjusted to the domain boundary). In this novel method, we only collocate on nodes interior to the domain boundary and enforce boundary conditions as constraints by means of Lagrange multipliers. This combination enables full geometric flexibility near boundaries without compromising the high-order accuracy of the RBF-FD method. The high-order accuracy and robustness of two formulations of this approach are illustrated by numerical experiments.