Non-Isotropic Jacobi Pseudospectral Method

The main advantage of spectral method is its high accuracy. However, this merit is de-stroyed oftentimes by singularities of genuine solutions, which could be caused by several facts,such as degenerating coe?cients of leading terms in di?erential equations. Moreover, the co-e?cients of derivatives of di?erent orders involved in underlying problems may degeneratein completely di?erent way. For solving such problems, Guo Ben-yu developed the Jacobiapproximation in certain non-uniformly Jacobi-weighted Sobolev space, and proposed the cor-responding Jacobi spectral method with its applications to one-dimensional singular di?erentialequations. The Jacobi spectral method is also very useful for many kinds of other related prob-lems, for examples, di?erential equations on unbounded domains and axisymmetric domains.On the other hand, some results on the Jacobi approximation have been successfully appliedto the analysis of various rational spectral methods.In practice, it is more important and interesting to solve multiple-dimensional singularproblems and related problems numerically. Guo Ben-yu and Wang Li-lian provided the Jacobispectral method in two-dimensions. Whereas, the pseudospectral method is more preferablein actual computation, since it only needs to evaluate unknown functions at interpolationnodes. This feature simplifies calculation and saves a lot of work. Furthermore, it is mucheasier to deal with nonlinear terms. Guo Ben-yu and Wang Li-lian investigated the Jacobipseudospectral method for one-dimensional singular problems. But, so far, there is no existingwork concerning the Jacobi pseudospectral method in multiple dimensions.This theses is devoted to the Jacobi pseudospectral method in multiple dimensions and itsapplications. In the first chapter, we recall the history of the related work and present the mainresults of this work. In chapter 2, we list and renew some basic results on the one-dimensionalJacobi approximation. In chapter 3, we establish the main results on the Jacobi-Gauss typeinterpolation in multiple-dimensional space, which play important role in designing and ana-lyzing various Jacobi pseudospectral schemes for singular problems and other related problems.We also derive a series of sharp results on the Legendre-Gauss type interpolation and the re-lated Bernstein-Jackson type inequalities, which are very useful for pseudospectral method ofpartial di?erential equations with non-constant coe?cients. As examples of applications, weconsider a two-dimensional singular problem in chapter 4, and a problem defined on an ax-isymmetric domain in chapter 5. The convergence of proposed schemes are proved. Numericalresults show the e?ciency of this new approach and agree with the theoretical analysis well.The final chapter is for some concluding remarks.