Accurate Computation of Laplace Eigenvalues by an Analytical Level Set Method

2012/03/24 by Grinfeld, Pavel
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Numerical Analysis (math.NA)

paper · doi:10.48550/arxiv.1203.5444

Abstract

This purpose of this write-up is to share an idea for accurate computation of Laplace eigenvalues on a broad class of smooth domains. We represent the eigenfunction u as a linear combination of eigenfunctions corresponding to the common eigenvalue ρ²:\EQN610\RD\CELLu(r,θ) =∑_n=0^NP_nJ_n(ρ) cos nθ,1We adjust the coefficients P_n and the parameter ρ so that the zero level set of u approximates the domain of interest. For some domains, such as ellipses of modest eccentricity, the coefficients P_n decay exponentially and the proposed method can be used to compute eigenvalues with arbitrarily high accuracy.

Related

• The fundamental Laplacian eigenvalue of the ellipse with Dirichlet\n boundary conditions
• Uniform level set estimates for ground state eigenfunctions
• Stable Computation of Laplacian Eigenfunctions Corresponding to Clustered Eigenvalues
• Computing ultra-precise eigenvalues of the Laplacian within polygons
• Numerical approximation of the first p-Laplace eigenpair