Year: 2012
Type: preprint
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 $ρ^{2}$:\EQN{6}{1}{}{0}{\RD{\CELL{u(r,θ) =\sum_{n=0}^{N}P_{n}J_{n}(ρ) \cos nθ,}}{1}{}{}{}}We 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.
Source: arXiv (Cornell University)
Author: Grinfeld, Pavel
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W1867443980