Tensor solutions in irregular domains: Eigenvalue problems

Abstract

This work develops a numerical method to solve eigenvalue problems considering irregular two-dimension convex domains. It is based on the application of the Jordan spectral method to domains sectionally delimited by C1 class convex Jordan curves. The approach is straightforward in Cartesian coordinates and eliminates the use of polar coordinates or conformal maps, revealing an expressive accuracy gain compared to finite differences approach, obtaining solutions with more than ten digits of precision. Being computer-oriented and presenting spectral accuracy, the method is effective for several geometries, allowing to calculate margins of error in eigenvalues and eigenfunctions, being able to identify spurious results. Besides, the approach can be applied to either Dirichlet or Neumann boundary conditions. Helmholtz equation is used as an example, but the methodology is general and can be applied to any other eigenvalue partial differential equation. The approach does not assume a priori symmetry considerations, which allows the emergence of solutions to non-symmetrical vibration modes.