Novel approach to spectral methods for irregular domains

Abstract

Herein, the solution of partial differential equations (PDEs) using spectral methods is developed for irregular domains, which preserves their accuracy. Previously, to solve these problems, the finite differences method or the embedded domains method was typically applied. The approach presented in this article can be used for any boundary described by a Jordan curve, and the solution behavior outside the domain need not to considered. The computational process has low cost and generality because the map constructions and changing variables are unnecessary. In addition, by using the presented parametrization process, boundary conditions (boundary bordering) can be implemented conveniently, where the rectangular domains can be considered as an asymptotical case. The structure is numerically oriented, which facilitates the application of any algorithm related to spectral methods.