Error analysis of spectral method on a triangle
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Error analysis of spectral method on a triangle Ben-yu Guo a,b, and Li-Lian Wang c a Department of Mathematics, Shanghai Normal University, Shanghai, 200234, China
E-mail: [email protected] b Division of Computational Science of E-institute of Shanghai Universities c Department of Mathematics, Nan yang Technology University, 637616, Singapore
Received 2 July 2004; accepted 28 December 2004 Communicated by Z.Y. Chen
In this paper, the orthogonal polynomial approximation on triangle, proposed by Dubiner, is studied. Some approximation results are established in certain non-uniformly Jacobiweighted Sobolev space, which play important role in numerical analysis of spectral and triangle spectral element methods for differential equations on complex geometries. As an example, a model problem is considered. Keywords: spectral method on triangle, convergence Mathematics subject classifications (2000): 33C45, 41A10, 41A25, 65N35
1.
Introduction
Although spectral methods have gained increasingly popularity in scientific computations during the last 30 years, its applications to problems on complex geometries have been historically limited. Indeed, the standard spectral methods are traditionally confined to problems on regular domains. However, in many areas, the underlying problems are originally set on some complex domains, which usually require the use of numerical methods on irregular meshes. Consequently, the low-order finite element method and finite volume method are preferable in practice, since they allow geometric flexibility. Recently, high-order methods have become popular in computational fluid dynamics, for instance, the viscous flow equations around complex obstacles (cf. [16]). The h − p finite element method and spectral element method are notable among the high-order methods. The work of this author is supported in part by NSF of China, N.10471095, Science Foundation of
Shanghai, N. 04JC14062, The Special Funds for Doctorial Authorities of Education Ministry of China, N. 20040270002, E-institutes of Shanghai Municipal Education Commission, N.E03004, The Shanghai Leading Academic Discipline Project N. T0401 and The Fund N.04DB15 of Shanghai Education Commission.
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B.-y. Guo, L.-L. Wang / Spectral method on a triangle
Dubiner [7] considered a system of polynomials derived from Jacobi polynomials: gl,m (x, y) := 2l+3/2 (1−y)l Jl(0,0) (ξ )Jm(2l+1,0) (η),
ξ=
2x + y − 1 , η = 2y−1, (1.1) 1−y
which are L2 (T )-orthogonal on the triangle T := (x, y): 0 x 1, 0 y 1, 0 x + y 1 .
(1.2)
Later, Cai [5] extended this basis to the basis functions on curve surface for numerical simulation of electromagnetic scattering. The orthogonal polynomials (1.1) were also used as effective basis functions in h − p finite element method and spectral finite element method in [16,18,19], which exhibit geometric flexibility and spatial accuracy of high order in actual computations. There have been also other families of orthogonal polynomials on triangle. For instance, Appell and de Férie
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