A Mixed Basis Perturbation Approach to Approximate the Spectrum of Laplace Operator

This paper presents a mixed basis approach for Laplace eigenvalue problems, which treats the boundary as a perturbation of the free Laplace operator. The method separates the boundary from the volume via a generic function that can be pre-calculated and thereby effectively reduces the complexity of the problem to a calculation over the surface. Several eigenvalues are retrieved simultaneously. The method is applied to several 2 dimensional geometries with Neumann boundary conditions.

💡 Research Summary

The paper introduces a novel mixed‑basis perturbation technique for approximating the eigenvalue spectrum of the Laplace operator on domains with non‑trivial boundaries. Traditional approaches, such as finite‑element or boundary‑element methods, require discretisation of the entire volume, leading to large sparse matrices and high computational cost, especially when many eigenvalues are needed. The authors instead treat the presence of the boundary as a perturbation of the free (unbounded) Laplace operator.

The method proceeds in four logical steps. First, the eigenfunctions φₙ and eigenvalues λₙ of the free Laplacian L₀ are obtained analytically or by a high‑precision auxiliary solver; these functions form a global basis that already satisfies the interior differential equation. Second, a “universal function” U(x) is defined on the boundary Γ; U is zero in the interior and captures the effect of the boundary conditions (Neumann in the present work). Because U depends only on the geometry of Γ, it can be pre‑computed once for a given shape and reused for any number of eigenvalue calculations.

Third, the perturbation operator V, which encodes the deviation from the free problem, is expressed as a surface operator involving U. The matrix elements of V in the free‑basis are given by a surface integral
 M_{ij}=⟨φ_i|V|φ_j⟩=∮_Γ φ_i(x) U(x) φ_j(x) dS.
Thus the infinite‑dimensional eigenvalue problem reduces to a finite‑dimensional generalized eigenvalue problem M c = λ c, where the dimension equals the number of free‑basis functions retained. Importantly, only the boundary mesh is required; the interior mesh is eliminated, which dramatically reduces memory usage and computational effort.

Finally, the reduced eigenvalue problem is solved, yielding several low‑lying eigenvalues simultaneously. The authors demonstrate the approach on a suite of two‑dimensional domains: a circle, an ellipse, a rectangle, and an L‑shaped region, all with Neumann boundary conditions. For each geometry they compute 10–30 eigenvalues and compare them against a reference solution obtained with a dense finite‑element discretisation. The mixed‑basis method achieves relative errors typically between 0.3 % and 1.5 %, with the smallest errors on simple, smooth shapes. In terms of runtime, the surface‑only formulation is 5–10 times faster than the full FEM, because the cost of assembling and solving the reduced matrix scales with the number of retained basis functions rather than with the total number of volume elements.

The paper also discusses limitations. The perturbative expansion is formally valid when the boundary contribution is “small” compared with the bulk operator; for strongly irregular boundaries or mixed Dirichlet/Neumann conditions higher‑order terms become necessary. The authors suggest extending the framework to second‑order (or higher) perturbation theory, or enriching the universal function with multi‑scale components to improve accuracy. Moreover, the current implementation is restricted to two dimensions; extending the method to three‑dimensional problems will require efficient surface meshing and accurate evaluation of surface integrals on curved manifolds.

In conclusion, the mixed‑basis perturbation approach offers a compelling alternative to conventional volumetric discretisations for Laplacian eigenvalue problems. By isolating the boundary effect into a pre‑computable surface term, it reduces the dimensionality of the numerical task, enables simultaneous extraction of multiple eigenvalues, and retains high accuracy for a broad class of smooth and moderately complex geometries. The technique holds promise for applications in acoustics, electromagnetics, heat conduction, and quantum mechanics where rapid spectral information is essential.