Spectral Element Method And Domain Decomposition Pseudospectral Method For Mixed Inhomogeneous Boundary Value Problems Of Partial Differential Equations On Polygons

During the past three decades, spectral and pseudospectral methods developed rapidly, which serve as important tools for solving differential equations numerically. The fascinating merit of spectral and pseudospectral methods is their high accuracy. In other words, the smoother the exact solutions, the smaller the errors of numerical solutions. In the early work, we proposed the numerical methods based on the fast convergence of Fourier, Legendre and Chebyshev orthogonal approximations. Later, we studied various spectral and pseudospectral methods for degenerated problems, singular problems and some problems defined on unbounded domains, based on the nice features of Jacobi, Laguerre and Hermite orthogonal approximations. However, the standard Jacobi, Laguerre and Hermite spectral and pseudospectral methods are traditionally confined to problems defined on or outside rectangular domains. This fact is the main disadvantage of existing spectral and pseudospectral methods. Recently, in order to develop the spectral and pseudospectral methods, some authors focus on new approaches, and try to remove the above serious drawback and enlarge the applications of spectral and pseudospectral methods.In this thesis, we investigate spectral and pseudospectral methods for mixed inhomogeneous boundary value problems on quadrilaterals, as well as spectral element and domain decomposition pseudospectral methods for mixed inhomogeneous boundary value problems defined on polygons.We first introduce a family of functions, which are induced by the Legendre polynomials and certain proper variable transformation. They form a complete orthogonal system on a convex quadrilateral ?. The corresponding results on the L~2 (Ω)-orthogonal approximation and the H_O~1(Ω)- orthogonal approximation are established, which play important roles in spectral method on quadrilaterals. We also design the spectral schemes for a Dirichlet boundary value problem of Poisson equation and an initial-boundary value problem of parabolic equation, respectively. Numerical results show the high efficiency of our new algorithms.We next introduce a new family of functions as the base functions, and establish the basic results on the corresponding orthogonal and quasi-orthogonal approximations on quadrilaterals. Then, we propose a composite quasi-orthogonal approximation on a polygon, which matches the numerical solutions on the common boundaries of adjacent subdomains and keeps the global spectral accuracy on the polygons. As important applications, we consider the Petrov-Galerkin spectral method for a mixed inhomogeneous boundary value problem defined on quadrilaterals, and the Petrov-Galerkin spectral element method for polygons. We prove their spectral accuracy.Thirdly, we introduce an another family of functions induced by the Legendre polynomials as the base functions and establish the basic results on the related Legendre-Gauss type interpolation on quadrilaterals. We also build up the corresponding numerical quadratures, which keep the exactness for the above base functions and their derivatives of first order. Then, we provide the spectral schemes for two model problems, and prove their convergence and the spectral accuracy in space. Numerical results coincide well with the theoretical analysis .Finally, we consider the quasi-orthogonal approximation on quadrilaterals and polygons, and the related Legendre-Gauss type interpolations. We derive the basic results on these approximations. Then, we design the corresponding numerical quadratures, which keep the exactness for the base functions and their derivatives of first order. We also propose domain decomposition pseudospectral method for a mixed inhomogeneous boundary value problem on polygons, and prove the global spectral accuracy of this new approach.This thesis overcomes the main shortcoming of the usual spectral and pseudospectral methods. The approximation results and techniques in this work are applicable to arbitrary polygons. Therefore, this work enriches the theory of spectral and pseudospectral methods, and enlarges their applications essentially.