Mesh-free numerical method for Dirichlet eigenpairs of the Laplacian with potential

This paper is concerned with the numerical approximation of the $L^2$ Dirichlet eigenpairs of the operator $-Δ+ V$ on a simply connected $C^2$ bounded domain $Ω\subset \mathbb{R}^2$ containing the origin, where $V$ is a radial potential. We propose a mesh-free method inspired by the Method of Particular Solutions for the Laplacian (i.e. $V=0$). Extending this approach to general $C^1$ radial potentials is challenging due to the lack of explicit basis functions analogous to Bessel functions. To overcome this difficulty, we consider the equation $-Δu + V u = λu$ on a ball containing $Ω$, without imposing boundary conditions, for a collection of values $λ$ forming a fine discretisation of the interval in which eigenvalues are sought. By rewriting the problem in polar coordinates and applying a Fourier expansion with respect to the angular variable, we obtain a decoupled system of ordinary differential equations. These equations are solved numerically using a one-dimensional Finite Element Method, yielding a family of basis functions that are solutions of the equation $-Δu + V u = λu$ on the ball and are independent of the domain $Ω$. Dirichlet eigenvalues of $-Δ+ V$ are then approximated by minimising the boundary values on $\partial Ω$ among linear combinations of the basis functions and identifying those values of $λ$ for which the computed minimum is sufficiently small. The proposed method is highly memory-efficient compared to the standard Finite Element approach.

💡 Research Summary

The paper introduces a mesh‑free algorithm for computing Dirichlet eigenpairs of the operator (-\Delta+V) on a simply‑connected, (C^{2}) bounded domain (\Omega\subset\mathbb{R}^{2}) that contains the origin, where the potential (V) is a radial, (C^{1}) function. Classical approaches rely on triangulating (\Omega) and solving a large generalized eigenvalue problem via finite‑element (FEM) discretisation; such methods generate matrices whose size scales like (h^{-4}) (with mesh size (h)), leading to severe memory and computational constraints, especially when the domain changes.

The authors adapt the Method of Particular Solutions (MPS), originally devised for the pure Laplacian ((V=0)), to the more general case with an arbitrary radial potential. The key obstacle is the lack of explicit analytic basis functions (e.g., Bessel functions) for (-\Delta+V). To bypass this, they embed (\Omega) in a larger ball (B_{O}(R)) and, for a dense set of trial spectral parameters (\lambda\in