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A new triangular spectral element method. I: Implementation and analysis on a triangle. (English) Zbl 1280.65131
Authors’ abstract: This paper serves as our first effort to develop a new triangular spectral element method (TSEM) on unstructured meshes, using the rectangle-triangle mapping proposed in the conference note [Y. Li et al., Lect. Notes Comput. Sci. Engeg. 76, 237–246 (2011; Zbl 1216.65169)]. Here, we provide some new insights into the originality and distinctive features of the mapping, and show that this transform only induces a logarithmic singularity, which allows us to devise a fast, stable and accurate numerical algorithm for its removal. Consequently, any triangular element can be treated as efficiently as a quadrilateral element, which affords a great flexibility in handling complex computational domains. Benefitting from the fact that the image of the mapping includes the polynomial space as a subset, we are able to obtain optimal $$L^2$$- and $$H^1$$-estimates of approximation by the proposed basis functions on triangle. The implementation details and some numerical examples are provided to validate the efficiency and accuracy of the proposed method. All these will pave the way for developing an unstructured TSEM based on, e.g., the hybridizable discontinuous Galerkin formulation.

##### MSC:
 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 65N35 Spectral, collocation and related methods for boundary value problems involving PDEs 65N22 Numerical solution of discretized equations for boundary value problems involving PDEs 65F05 Direct numerical methods for linear systems and matrix inversion 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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##### References:
 [1] Adams, R.A.: Sobolev Spaces. Acadmic Press, New York (1975) · Zbl 0314.46030 [2] Boyd, JP; Yu, F, Comparing seven spectral methods for interpolation and for solving the Poisson equation in a disk: Zernike polynomials, logan-Shepp ridge polynomials, Chebyshev-Fourier series, cylindrical robert functions, Bessel-Fourier expansions, square-to-disk conformal mapping and radial basis functions, J. Comput. Phys., 230, 1408-1438, (2011) · Zbl 1210.65192 [3] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Fundamentals in Single Domains. Scientific Computation. Springer-Verlag, Berlin (2006) · Zbl 1093.76002 [4] Canuto, C., Hussaini, M.Y., Quarteroni, A., Zang, T.A.: Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics. Scientific Computation. Springer, Berlin (2007) · Zbl 1121.76001 [5] Chen, L; Shen, J; Xu, C, A triangular spectral method for the Stokes equations, Numer. Math.: Theory Methods Appl., 4, 158-179, (2011) · Zbl 1249.65262 [6] Chen, Q; Babuška, IM, Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle, Comput. Methods Appl. Math. Eng., 128, 405-417, (1995) · Zbl 0862.65006 [7] Chernov, A, Optimal convergence estimates for the trace of the polynomial $$L$$\^{}{2}-projection operator on a simplex, Math. Comput., 81, 765-787, (2011) · Zbl 1242.41007 [8] Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North Holland, Amsterdam, The Netherlands (1978) · Zbl 0383.65058 [9] Cockburn, B; Gopalakrishnan, J; Lazarov, R, Unified hybridization of discontinuous Galerkin, mixed, and continuous Galerkin methods for second order elliptic problems, SIAM J. Numer. Anal., 47, 1319-1365, (2009) · Zbl 1205.65312 [10] Deville, M.O., Fischer, P.F., Mund, E.H.: High-Order Methods for Incompressible Fluid Flow. Cambridge Monographs on Applied and Computational Mathematics, vol. 9. Cambridge University Press, Cambridge (2002) · Zbl 1007.76001 [11] Dubiner, M, Spectral methods on triangles and other domains, J. Sci. Comput., 6, 345-390, (1991) · Zbl 0742.76059 [12] Duffy, MG, Quadrature over a pyramid or cube of integrands with a singularity at a vertex, SIAM J. Numer. Anal., 19, 1260-1262, (1982) · Zbl 0493.65011 [13] Gautschi, W, Gauss quadrature routines for two classes of logarithmic weight functions, Numer. Algorithms, 55, 265-277, (2010) · Zbl 1200.65019 [14] Gordon, WJ; Hall, CA, Construction of curvilinear co-ordinate systems and applications to mesh generation, Int. J. Numer. Methods Eng., 7, 461-477, (1973) · Zbl 0271.65062 [15] Gradshteyn, I.S., Ryzhik, I.M.: Table of Integrals, Series and Products, 7th Edn. Academic Press, New York (2007) · Zbl 1208.65001 [16] Guo, BY; Shen, J; Wang, L, Optimal spectral-Galerkin methods using generalized Jacobi polynomials, J. Sci. Comput, 27, 305-322, (2006) · Zbl 1102.76047 [17] Guo, BY; Wang, L, Error analysis of spectral method on a triangle, Adv. Comput. Math., 26, 473-496, (2007) · Zbl 1116.65122 [18] Heinrichs, W, Spectral collocation schemes on the unit disc, J. Comput. Phys., 199, 55-86, (2004) · Zbl 1057.65089 [19] Helenbrook, BT, On the existence of explicit hp-finite element methods using Gauss-lobatto integration on the triangle, SIAM J. Numer. Anal., 47, 1304-1318, (2009) · Zbl 1191.65154 [20] Hesthaven, JS, From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. Numer. Anal., 35, 655-676, (1998) · Zbl 0933.41004 [21] Hylleraas, EA, Linearization of products of Jacobi polynomials, Math. Scand., 10, 189-200, (1962) · Zbl 0109.29603 [22] Karniadakis, G.E., Sherwin, S.J.: Spectral/hp Element Methods for Computational Fluid Dynamics. Numerical Mathematics and Scientific Computation, 2nd Edn. Oxford University Press, New York (2005) · Zbl 1116.76002 [23] Kirby, RM; Sherwin, SJ; Cockburn, B, To CG or to HDG: a comparative study, J. Sci. Comput., 51, 183-212, (2012) · Zbl 1244.65174 [24] Koornwinder, T.: Two-Variable Analogues of the Classical Orthogonal Polynomials. In: Theory and Application of Special Functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) pp. 435-495. Math. Res. Center, Univ. Wisconsin, Publ. No, p. 35. Academic Press, New York (1975) · Zbl 0326.33002 [25] Li, H; Shen, J, Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle, Math. Comput., 79, 1621-1646, (2010) · Zbl 1197.65176 [26] Li, H; Wang, L, A spectral method on tetrahedra using rational basis functions, Int. J. Numer. Anal. Model., 7, 330-355, (2010) [27] Li, Y; Wang, L; Li, H; Ma, H, A new spectral method on triangles, No. 76, 237-246, (2011), New York · Zbl 1216.65169 [28] Nguyen, NC; Peraire, J; Cockburn, B, Hybridizable discontinuous Galerkin methods, No. 76, 63-84, (2011), New York · Zbl 1216.65160 [29] Pasquetti, R; Rapetti, F, Spectral element methods on unstructured meshes: comparisons and recent advances, J. Sci. Comput., 27, 377-387, (2006) · Zbl 1102.65119 [30] Pasquetti, R; Rapetti, F, Spectral element methods on unstructured meshes: which interpolation points?, Numer. Algorithms, 55, 349-366, (2010) · Zbl 1200.65096 [31] Patera, AT, A spectral element method for fluid dynamics: laminar flow in a channel expansion, J. Comput. Phys., 54, 468-488, (1984) · Zbl 0535.76035 [32] Schwab, C.: $$p$$- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Numerical Mathematics and Scientific Computation. Oxford Science, Oxford, UK (1998) 587 · Zbl 0910.73003 [33] Shen, J., Tang, T., Wang, L.: Spectral Methods: Algorithms, Analysis and Applications. Springer Series in Computational Mathematics, Vol. 41. Springer-Verlag, Berlin Heidelberg (2011) · Zbl 1227.65117 [34] Shen, J; Wang, L; Li, H, A triangular spectral element method using fully tensorial rational basis functions, SIAM J. Numer. Anal., 47, 1619-1650, (2009) · Zbl 1197.65187 [35] Szegö, G.: Orthogonal Polynomials, 4th Edn, Vol. 23. AMS Coll. Publ., Providence, RI (1975) · JFM 61.0386.03 [36] Taylor, MA; Wingate, BA; Vincent, RE, An algorithm for computing Fekete points in the triangle, SIAM J. Numer. Anal., 38, 1707-1720, (2000) · Zbl 0986.65017 [37] Weber, H.: Lehrbuch der Algebra. Erster Band, Braunschweig (1912) · JFM 43.0143.01 [38] Xie, Z., Wang, L., Zhao, X.: On exponential convergence of Gegenbauer interpolation and spectral differentiation. Math. Comput., electronically published on 21 August 2012 · Zbl 1262.30023 [39] Xu, Y.: Common Zeros of Polynomials in Several Variables and Higher Dimensional Quadrature. Chapman & Hall/CRC, London, UK (1994) · Zbl 0898.26004 [40] Xu, Y, On Gauss-lobatto integration on the triangle, SIAM J. Numer. Anal., 49, 541-548, (2011) · Zbl 1221.65083
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