A multi-domain spectral collocation method for PDEs in curved domains with holes

Abstract

This paper presents a novel spectral collocation method that combines domain decomposition and mapping techniques for solving second-order elliptic equations with variable coefficients, as well as time-dependent advection–diffusion–reaction equations in a curved domain with holes. The process begins by partitioning the curved domain with holes into several subdomains. Each subdomain is then mapped to a regular domain through a polar coordinate transformation. Following this, linear transformations are applied to map these subdomains onto the reference element $(-1,1)\times (-1,1)$. Numerical simulations are performed on each reference element using the classical spectral collocation method. The numerical results demonstrate the high accuracy of the proposed approach.