Accurate Computation of Laplace Eigenvalues by an Analytical Level Set Method
This method offers a novel approach to accurately compute Laplace eigenvalues for a class of smooth domains, which is important for applications in physics and engineering.
The paper presents an analytical level set method for computing Laplace eigenvalues on smooth domains by representing eigenfunctions as linear combinations of Bessel functions, achieving arbitrarily high accuracy for domains like ellipses with modest eccentricity.
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.