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

Morten E. Nielsen, Bengt Fornberg

TL;DR

The paper introduces a high-order, meshfree approach for solving elliptic PDEs on unfitted node sets by coupling radial basis function-generated finite differences (RBF-FD) with Lagrange multiplier boundary constraints. By collocating only interior nodes and enforcing Dirichlet/Neumann conditions as equality constraints, two formulations (RBF-FD-LM1 and RBF-FD-LM2) achieve high-order accuracy while preserving geometric flexibility, with convergence up to $\mathcal{O}(h^{10})$. Numerical experiments on 2D and 3D Poisson problems with complex domains demonstrate that the LM formulations can match or surpass traditional RBF-FD-C accuracy, particularly when using unfitted node sets, and reveal the robustness of the LM2 formulation across test cases. The work highlights a practical, scalable path for precise boundary handling in meshfree methods and suggests potential extensions to moving boundaries and time-dependent problems.

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.

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

TL;DR

The paper introduces a high-order, meshfree approach for solving elliptic PDEs on unfitted node sets by coupling radial basis function-generated finite differences (RBF-FD) with Lagrange multiplier boundary constraints. By collocating only interior nodes and enforcing Dirichlet/Neumann conditions as equality constraints, two formulations (RBF-FD-LM1 and RBF-FD-LM2) achieve high-order accuracy while preserving geometric flexibility, with convergence up to . Numerical experiments on 2D and 3D Poisson problems with complex domains demonstrate that the LM formulations can match or surpass traditional RBF-FD-C accuracy, particularly when using unfitted node sets, and reveal the robustness of the LM2 formulation across test cases. The work highlights a practical, scalable path for precise boundary handling in meshfree methods and suggests potential extensions to moving boundaries and time-dependent problems.

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.
Paper Structure (15 sections, 2 theorems, 22 equations, 14 figures)

This paper contains 15 sections, 2 theorems, 22 equations, 14 figures.

Table of Contents

  1. Introduction
  2. Methodology
  3. Radial basis function-generated finite differences
  4. RBF-FD-C method
  5. Lagrange multiplier-based RBF-FD methods
  6. Lagrange multiplier method for equality-constrained linear systems
  7. RBF-FD-LM - Formulation 1
  8. RBF-FD-LM - Formulation 2
  9. Comments on the RBF-FD-LM methods
  10. Numerical experiments
  11. Test problem 1: Poisson's equation in the unit disk with Dirichlet boundary conditions
  12. Test problem 2: Poisson's equation in butterfly domain with Dirichlet and Neumann boundary conditions
  13. Test problem 3: Poisson's equation in the sphere with Dirichlet boundary conditions
  14. Test problem 4: Poisson's equation in the sphere with Dirichlet and Neumann boundary conditions
  15. Conclusions

Key Result

Theorem 2.1

\newlabeltheo:t10 The vector $\bm{x}$ that for (eq:l2min) minimizes $J(\bm{x}) = \frac{1}{2} || A\bm{x} - \bm{b}||^2_2 = \frac{1}{2}(A\bm{x}-\bm{b})^T (A\bm{x}-\bm{b})$ can be obtained by solving the square linear system,

Figures (14)

  • Figure 1: Domain with boundary definitions for $d = 2$.
  • Figure 1: Boundary-fitted node set including definitions of exterior (gray dots), Dirichlet (red circles), Neumann (blue squares) and interior (black dots) boundary nodes. This node set combines quasi-uniformity with nodes being boundary-fitted.
  • Figure 1: The three different node sets and collocation setups used for test problem 1. The stencil illustrated on each node set is based on a relative stencil size $\lceil n/\ell \rceil = 2$ and augmented polynomial degree $m = 2$.
  • Figure 2: Unfitted node set including definitions of exterior (gray dots), Dirichlet (red circles), Neumann (blue squares) and interior (black dots) boundary nodes. The interior nodes are distributed with no respect to the boundary nodes.
  • Figure 2: $\text{Log}_{10}$ of the relative $\ell_2$-norm errors as function of relative stencil size ($\lceil n/\ell \rceil$) and augmented polynomial degree ($m$) using the radial basis function $\phi(r) = r^3$ on a boundary-fitted node set, where all three methods use interior and boundary nodes as collocation nodes, for solving the Poisson equation in the unit disk with Dirichlet boundary conditions.
  • ...and 9 more figures

Theorems & Definitions (2)

  • Theorem 2.1
  • Theorem 2.2