Well-Conditioning Map for Tensor Spectral Solutions for Eigenvalues in 2D Irregular Planar Domains

Abstract

This study proposes a diffeomorphic map for planar irregular domains, which transforms them into a unitary circle suitable for implementing spectral solutions on a complete and orthogonal basis. The domain must be enclosed by a Jordan curve that is univocally described in polar coordinates by a $C^2$ class function. Because of the achieved precision, general eigenvalue problems can be solved, yielding hundreds of eigenvalues and eigenfunctions in a single shot with at least ten-digit precision. In a constructive approach, there is no need for special functions or roots. The examples focus on the Helmholtz equation, but the method applies to other eigenvalue problems such as Hamiltonians. For validation, we compare all the results with the values obtained by finite elements (FE), some with other studies, and discuss sharp error estimate criteria and the asymptotic behavior of the eigenvalues. To demonstrate the generality of the approach, we present several examples with different geometries in which solutions emerge naturally. Considering the spectral accuracy, half of the examples are unpublished solutions, and the other half agree with or expand the precision of previous studies using different approaches.