Optimal Spectral-Galerkin Methods Using Generalized Jacobi Polynomials

Authors:
Ben-Yu Guo;Jie Shen;Li-Lian Wang
Affiliations:
Department of Mathematics, Shanghai Normal University and Shanghai E-Institute for Computational Sciences, Shanghai, P. R. China 200234;Department of Mathematics, Purdue University, West Lafayette, USA 47907;Aff2 Aff3
Venue:
Journal of Scientific Computing
Year:
2006

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Abstract

We extend the definition of the classical Jacobi polynomials withindexes 驴, β驴1 to allow 驴 and/or β to be negative integers. We show that the generalized Jacobi polynomials, with indexes corresponding to the number of boundary conditions in a given partial differential equation, are the natural basis functions for the spectral approximation of this partial differential equation. Moreover, the use of generalized Jacobi polynomials leads to much simplified analysis, more precise error estimates and well conditioned algorithms.