Composite Spectral Method for Exterior Problems with Polygonal Obstacles

In this paper, we propose a domain decomposition spectral method for exterior problems with arbitrary polygonal obstacles. Some results on the composite Legendre-Laguerre quasi-orthogonal approximation are established, which play important roles in the spectral method for exterior problems. As examples of applications, the composite spectral schemes are provided for two model problems, with the convergence analysis. Numerical results demonstrate the spectral accuracy of this new approach. The approximation results and techniques developed in this paper are also applicable to other problems defined on unbounded domains with complex geometry.