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

Dragoş Manea

TL;DR

The article addresses computing $L^2$ Dirichlet eigenpairs of the Schrödinger-type operator $-\Delta+V$ with a radial potential on planar domains by a mesh-free strategy inspired by the Method of Particular Solutions. It extends classical MPS ideas to general radial potentials by solving decoupled angular modes on an enclosing ball with a 1D FEM for the radial part, generating basis functions independent of the domain. Eigenpairs are recovered by minimizing the boundary-to-domain $L^2$-norm quotient $\mathcal{F}^{\lambda}$ over a discretized spectral parameter $\lambda$, with rigorous convergence and no-missed-eigenpair guarantees. The approach yields substantial memory savings and competitive accuracy, demonstrated on ellipsoidal and star-shaped domains, and offers a path toward higher dimensions and Neumann problems with further refinement.

Abstract

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.

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

TL;DR

The article addresses computing Dirichlet eigenpairs of the Schrödinger-type operator with a radial potential on planar domains by a mesh-free strategy inspired by the Method of Particular Solutions. It extends classical MPS ideas to general radial potentials by solving decoupled angular modes on an enclosing ball with a 1D FEM for the radial part, generating basis functions independent of the domain. Eigenpairs are recovered by minimizing the boundary-to-domain -norm quotient over a discretized spectral parameter , with rigorous convergence and no-missed-eigenpair guarantees. The approach yields substantial memory savings and competitive accuracy, demonstrated on ellipsoidal and star-shaped domains, and offers a path toward higher dimensions and Neumann problems with further refinement.

Abstract

This paper is concerned with the numerical approximation of the Dirichlet eigenpairs of the operator on a simply connected bounded domain containing the origin, where is a radial potential. We propose a mesh-free method inspired by the Method of Particular Solutions for the Laplacian (i.e. ). Extending this approach to general radial potentials is challenging due to the lack of explicit basis functions analogous to Bessel functions. To overcome this difficulty, we consider the equation on a ball containing , without imposing boundary conditions, for a collection of values 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 on the ball and are independent of the domain . Dirichlet eigenvalues of are then approximated by minimising the boundary values on among linear combinations of the basis functions and identifying those values of for which the computed minimum is sufficiently small. The proposed method is highly memory-efficient compared to the standard Finite Element approach.
Paper Structure (30 sections, 32 theorems, 285 equations, 4 figures, 2 tables)

This paper contains 30 sections, 32 theorems, 285 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction and main results
  2. Presentation of the problem
  3. Description of the method
  4. Our method converges: if the quotient $\mathcal{F}^\lambda({\boldsymbol{\alpha}})$ is small, then we are near an eigenpair
  5. No eigenpair is missed by the algorithm: the quotient $\mathcal{F}^\lambda({\boldsymbol{\alpha}})$ is small near every eigenpair
  6. Extension to a ball while preserving the PDE
  7. Fourier expansion with respect to the angular variable and truncation error estimates
  8. Function spaces for the Fourier coefficients
  9. Solving the weak problem to determine the Fourier coefficients
  10. Continuity in $\lambda$ for the Fourier coefficients
  11. Numerical approximation of the basis functions $\mathbf{u}_j^\lambda$ via one-dimensional Finite Element Method
  12. One-dimensional Finite Element spaces. Error estimates for the linear interpolation
  13. Quadrature for the term involving the potential $V$
  14. Approximating the Fourier coefficients using one-dimensional Finite Element Method
  15. Numerical approximation of the quotient $\mathcal{F}^\lambda({\boldsymbol{\alpha}})$
  16. ...and 15 more sections

Key Result

Theorem 1.1

Let $\Omega\subset B_O(R)$ be a simply connected $C^2$ bounded domain in $\mathbb{R}^2$ containing the origin $O$. Let $V\in C^1([0,R])$ be a potential taking values in $[1,\infty)$, and let $K>1$. Then, the following statements hold: Interpretation of statement item:main-II: if the truncation parameter $J$ is chosen sufficiently large and the discretisation of the interval $[1,K]$ is fine enough

Figures (4)

  • Figure 1: The function $\mathcal{F}^\lambda_h(\alpha_{min}^\lambda)$ in terms of the test value $\lambda$, for the ellipse $\Omega_E$ (A) and the star-shaped domain $\Omega_S$ (B).
  • Figure 2: Detailed view in the neighbourhood of a local minimum of the function $\mathcal{F}^\lambda_h(\alpha_{min}^\lambda)$ in terms of the test value $\lambda$, for the ellipse $\Omega_E$ (A) and the star-shaped domain $\Omega_S$ (B).
  • Figure 3: Approximation of the eigenfunction corresponding to the third eigenvalue in the case of the elliptic domain $\Omega_E$.
  • Figure 4: The approximations of two orthonormal eigenfunctions corresponding to the seventh eigenvalue in the case of the star-shaped domain $\Omega_S$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Theorem 3.1
  • Proposition 3.2: Runge property in the $H^1$ norm
  • proof
  • Proposition 3.3
  • Proposition 3.4
  • Proposition 3.5
  • Corollary 3.6
  • ...and 43 more