Accurate and Efficient Spectral Methods for Elliptic PDEs in Complex Domains

We develop accurate and efficient spectral methods for elliptic PDEs in complex domains using a fictitious domain approach. Two types of Petrov–Galerkin formulations with special trial and test functions are constructed, one is suitable only for the Poisson equation but with a rigorous error analysis, the other works for general elliptic equations but its analysis is not yet available. Our numerical examples demonstrate that our methods can achieve spectral convergence, i.e., the convergence rate only depends on the smoothness of the solution.