Irregular domains: Special coordinates for a pseudospectral method

Working with a special coordinate system, this study demonstrates how to obtain numerical solutions with geometric convergence for the eigenstates of a Laplacian operator in irregular prismatic domains (both annular and single) that are simply connected. An appropriate coordinate system, which defines a tightly bounded domain, allows for a fair mesh for series approximation nodes. Three independent criteria were used to verify the consistency of the solutions: the Rayleigh quotient, the divergence theorem, and a partial derivative equation (PDE) transformed from an eigenvalue problem to a boundary value problem with Robin conditions. Supporting the proposed method, examples show a few hundred eigenstates obtained in a single computation, with at least 10 significant figures and a low computational cost.