Mesh-Free Numerical Method for Dirichlet Eigenpairs of the Laplacian

Title: Mesh-Free Numerical Method for Dirichlet Eigenpairs of the Laplacian with Potential
Date: September 7 2026 (Monday Holiday) 10:00 am - Noon EDT
Summary: This paper is concerned with the numerical approximation of the L^2 Dirichlet eigenpairs of the operator -delta + V on a simply connected C^2 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.
Speaker: Dr. Dragoș Manea is a mathematician specialising in mathematical analysis and applied mathematics. He holds a Master of Science degree from the University of Oxford and completed his PhD in Mathematics in 2025. He is currently a Research Assistant at the “Simion Stoilow” Institute of Mathematics of the Romanian Academy in Bucharest, Romania. His doctoral research focused on the theoretical and numerical analysis of evolutionary partial differential equations, with particular emphasis on deriving asymptotic results. In addition, he is interested in the numerical treatment of elliptic inverse and eigenvalue problems using non-standard approaches that avoid explicit meshing of the computational domain. Beyond PDE theory, his work also explores optimisation methods and their applications across a wide spectrum of problems, ranging from theoretical questions — such as the asymptotic analysis of solutions to the Schrödinger equation — to practical engineering applications, including the optimisation of urban traffic and aircraft trajectories.
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