Non-isotropic Jacobi Spectral Method And Pseudospectral Method In Three Dimensions

In the past several decades, spectral methods as one of the important tools forscientific computing developed rapidly. They have been used widely for boundaryand initial-boundary value problems because of its high accuracy. The classic spectralmethods are available for periodic problems and various problems define onrectangular domains, by taking trigonometric polynomials, Legendre polynomials orChebysheve polynomials as the basis functions. But the high accuracy of numericalsolutions might be destroyed by the singularity of genuine solutions of the equations.In order to overcome this difficulty, we need develop Jacobi spectral method. Someauthors have investigated the Jacobi orthogonal projection and the Jacobi-Gaussinterpolation in non-uniformly Jacobi-weighted Sobolev spaces，and establishedsome approximate results on Jacobi projection and interpolation in one dimension.The Jacobi spectral method and pseudospectral method in two dimensions are alsoinvestigated by some authors in recently years. They have been successfully appliedto some singular problems and some problems define on unbounded oraxisymmetric domains.In this thesis, Non-isotropic Jacobi approximation in threedimensional non-uniformly weighted space are proposed, which form mathematicalfoundation of spectral method and pseudospectral method for singular problemsand problems define on unbounded or axisymmetric domains. Some results onJacobi approximation and Jacobi-Gauss type interpolation in three dimensions areestablished. The Jacobi spectral method and the Jacobi pseudospectral method areapplied to a three-dimensional singular problem. Meanwhile，the convergence ofthe proposed spectral scheme and pseudospectral scheme are proved. Numericalresults demonstrate the high efficiency of suggested algorithm.