COMPOSITE GENERALIZED LAGUERRE-LEGENDRE SPECTRAL METHOD WITH DOMAIN DECOMPOSITION AND ITS APPLICATION TO FOKKER-PLANCK EQUATION IN AN INFINITE CHANNEL

摘要

In this paper, we propose a composite generalized Laguerre-Legendre spectral method for partial differential equations on two-dimensional unbounded domains, which are not of standard types. Some approximation results are established, which are the mixed generalized Laguerre-Legendre approximations coupled with domain decomposition. These results play an important role in the related spectral methods. As an important application, the composite spectral scheme with domain decomposition is provided for the Fokker-Planck equation in an infinite channel. The convergence of the proposed scheme is proved. An efficient algorithm is described. Numerical results show the spectral accuracy in the space of this approach and coincide well with theoretical analysis. The approximation results and techniques developed in this paper are applicable to many other problems on unbounded domains. In particular, some quasi-orthogonal approximations are very appropriate for solving PDEs, which behave like parabolic equations in some directions, and behave like hyperbolic equations in other directions. They are also useful for various spectral methods with domain decompositions, and numerical simulations of exterior problems.