Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



The triangular spectral element method for Stokes eigenvalues

Authors: Weikun Shan and Huiyuan Li
Journal: Math. Comp. 86 (2017), 2579-2611
MSC (2010): Primary 65N35, 65N25, 35P15, 35Q30
Published electronically: March 29, 2017
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A triangular spectral element method is proposed for Stokes eigenvalues, which utilizes the generalized orthogonal Koornwinder polynomials as the local basis functions. The local polynomial projection, which serves as a Fortin interpolation on each triangular element, is defined by the truncated Koornwinder-Fourier series. A sharp estimate on the discrete inf-sup constant of the divergence for our triangular spectral element approximation scheme is then acquired via the stability analysis of the local projection operator. Further, the optimal error estimate of the $ H^1$-orthogonal spectral element projection oriented to Stokes equations is obtained through the globally continuous piecewise polynomial assembled by the union of all local projections. In the sequel, the optimal convergence rate/error estimate theory is eventually established for our triangular spectral element method for both eigenvalue and source problems of the Stokes equations. Finally, numerical experiments are presented to illustrate our theories on both the discrete inf-sup constant of the divergence and the accuracy of the computational eigenvalues.

References [Enhancements On Off] (What's this?)

  • [1] María G. Armentano and Verónica Moreno, A posteriori error estimates of stabilized low-order mixed finite elements for the Stokes eigenvalue problem, J. Comput. Appl. Math. 269 (2014), 132-149. MR 3197278,
  • [2] Mejdi Azaiez, Jie Shen, Chuanju Xu, and Qingqu Zhuang, A Laguerre-Legendre spectral method for the Stokes problem in a semi-infinite channel, SIAM J. Numer. Anal. 47 (2008/09), no. 1, 271-292. MR 2475939,
  • [3] Ivo Babuška, The finite element method with Lagrangian multipliers, Numer. Math. 20 (1972/73), 179-192. MR 0359352
  • [4] I. Babuška and B. Guo, Approximation properties of the $ h$-$ p$ version of the finite element method, Computer Methods in Applied Mechanics and Engineering 1133 (1996), no. 3-4, 319-346.
  • [5] I. Babuška and J. Osborn, Eigenvalue Problems, Handbook of Numerical Analysis, vol. 2, Elsevier, 1991, pp. 641-787.
  • [6] Ivo Babuška and Benqi Guo, Direct and inverse approximation theorems for the $ p$-version of the finite element method in the framework of weighted Besov spaces. II. Optimal rate of convergence of the $ p$-version finite element solutions, Math. Models Methods Appl. Sci. 12 (2002), no. 5, 689-719. MR 1909423,
  • [7] Christine Bernardi, Claudio Canuto, and Yvon Maday, Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem, SIAM J. Numer. Anal. 25 (1988), no. 6, 1237-1271. MR 972452,
  • [8] Christine Bernardi and Yvon Maday, Spectral methods, Handb. Numer. Anal., V, North-Holland, Amsterdam, 1997, pp. 209-485. MR 1470226,
  • [9] Sven Beuchler and Veronika Pillwein, Completions to sparse shape functions for triangular and tetrahedral $ p$-FEM, Domain decomposition methods in science and engineering XVII, Lect. Notes Comput. Sci. Eng., vol. 60, Springer, Berlin, 2008, pp. 435-442. MR 2436111,
  • [10] Sven Beuchler, Veronika Pillwein, Joachim Schöberl, and Sabine Zaglmayr, Sparsity optimized high order finite element functions on simplices, Numerical and Symbolic Scientific Computing, Texts Monogr. Symbol. Comput., Springer, New York and Vienna, 2012, pp. 21-44. MR 3060506,
  • [11] S. Beuchler and J. Schöberl, New shape functions for triangular $ p$-FEM using integrated Jacobi polynomials, Numer. Math. 103 (2006), no. 3, 339-366. MR 2221053,
  • [12] P. E. Bjørstad and B. P. Tjøstheim, High precision solutions of two fourth order eigenvalue problems, Computing 63 (1999), no. 2, 97-107. MR 1736662,
  • [13] J. M. Boland and R. A. Nicolaides, Stability of finite elements under divergence constraints, SIAM J. Numer. Anal. 20 (1983), no. 4, 722-731. MR 708453,
  • [14] John P. Boyd, Chebyshev and Fourier spectral methods, 2nd ed., Dover Publications, Inc., Mineola, NY, 2001. MR 1874071
  • [15] D. Braess, Finite Elements: Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 2007.
  • [16] James H. Bramble, A proof of the inf-sup condition for the Stokes equations on Lipschitz domains, Math. Models Methods Appl. Sci. 13 (2003), no. 3, 361-371. MR 1977631,
  • [17] F. Brezzi, On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 8 (1974), no. R-2, 129-151 (English, with loose French summary). MR 0365287
  • [18] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Scientific Computation, Springer-Verlag, Berlin, 2006. MR 2223552
  • [19] C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Evolution to Complex Geometries and Applications to Fluid Dynamics, Scientific Computation, Springer, Berlin, 2007. MR 2340254
  • [20] Wei Chen and Qun Lin, Approximation of an eigenvalue problem associated with the Stokes problem by the stream function-vorticity-pressure method, Appl. Math. 51 (2006), no. 1, 73-88. MR 2197324,
  • [21] Alexey Chernov, Optimal convergence estimates for the trace of the polynomial $ L^{2}$-projection operator on a simplex, Math. Comp. 81 (2012), no. 278, 765-787. MR 2869036,
  • [22] Lawrence K. Chilton, Locking-free mixed hp finite element methods for linear and geometrically nonlinear elasticity, ProQuest LLC, Ann Arbor, MI, 1997. Thesis (Ph.D.)-University of Maryland, Baltimore County. MR 2696632
  • [23] Charles F. Dunkl and Yuan Xu, Orthogonal Polynomials of Several Variables, Encyclopedia of Mathematics and its Applications, vol. 81, Cambridge University Press, Cambridge, 2001. MR 1827871
  • [24] Franco Brezzi and Michel Fortin, Mixed and Hybrid Finite Element Methods, Springer Series in Computational Mathematics, vol. 15, Springer-Verlag, New York, 1991. MR 1115205
  • [25] Benqi Guo and Ivo Babuška, Local Jacobi operators and applications to the $ p$-version of finite element method in two dimensions, SIAM J. Numer. Anal. 48 (2010), no. 1, 147-163. MR 2608363,
  • [26] Ben-Yu Guo and Li-Lian Wang, Error analysis of spectral method on a triangle, Adv. Comput. Math. 26 (2007), no. 4, 473-496. MR 2291668,
  • [27] Ben-Yu Guo, Jie Shen, and Li-Lian Wang, Optimal spectral-Galerkin methods using generalized Jacobi polynomials, J. Sci. Comput. 27 (2006), no. 1-3, 305-322. MR 2285783,
  • [28] Ben-Yu Guo, Jie Shen, and Li-Lian Wang, Generalized Jacobi polynomials/functions and their applications, Appl. Numer. Math. 59 (2009), no. 5, 1011-1028. MR 2495135,
  • [29] Jun Hu and Yunqing Huang, Lower bounds for eigenvalues of the Stokes operator, Adv. Appl. Math. Mech. 5 (2013), no. 1, 1-18. MR 3021142
  • [30] Pengzhan Huang, Lower and upper bounds of Stokes eigenvalue problem based on stabilized finite element methods, Calcolo 52 (2015), no. 1, 109-121. MR 3313590,
  • [31] ShangHui Jia, HongTao Chen, and HeHu Xie, A posteriori error estimator for eigenvalue problems by mixed finite element method, Sci. China Math. 56 (2013), no. 5, 887-900. MR 3047040,
  • [32] George Em Karniadakis and Spencer J. Sherwin, Spectral/$ hp$ Element Methods for Computational Fluid Dynamics, 2nd ed., Numerical Mathematics and Scientific Computation, Oxford University Press, New York, 2005. MR 2165335
  • [33] Tom Koornwinder, Two-variable analogues of the classical orthogonal polynomials, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Academic Press, New York, 1975, pp. 435-495. MR 0402146
  • [34] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, Second English edition, revised and enlarged. Translated from the Russian by Richard A. Silverman and John Chu. Mathematics and its Applications, Vol. 2, Gordon and Breach, Science Publishers, New York-London-Paris, 1969. MR 0254401
  • [35] R. B. Lehoucq, Implicitly restarted Arnoldi methods and subspace iteration, SIAM J. Matrix Anal. Appl. 23 (2001), no. 2, 551-562 (electronic). MR 1871329,
  • [36] Huiyuan Li and Jie Shen, Optimal error estimates in Jacobi-weighted Sobolev spaces for polynomial approximations on the triangle, Math. Comp. 79 (2010), no. 271, 1621-1646. MR 2630005,
  • [37] Huiyuan Li and Yuan Xu, Spectral approximation on the unit ball, SIAM J. Numer. Anal. 52 (2014), no. 6, 2647-2675. MR 3276427,
  • [38] Huipo Liu, Wei Gong, Shuanghu Wang, and Ningning Yan, Superconvergence and a posteriori error estimates for the Stokes eigenvalue problems, BIT 53 (2013), no. 3, 665-687. MR 3095260,
  • [39] Carlo Lovadina, Mikko Lyly, and Rolf Stenberg, A posteriori estimates for the Stokes eigenvalue problem, Numer. Methods Partial Differential Equations 25 (2009), no. 1, 244-257. MR 2473688,
  • [40] Yvon Maday, Dan Meiron, Anthony T. Patera, and Einar M. Rønquist, Analysis of iterative methods for the steady and unsteady Stokes problem: application to spectral element discretizations, SIAM J. Sci. Comput. 14 (1993), no. 2, 310-337. MR 1204233,
  • [41] Y. Maday, A. Patera, and E. Rønquist, A well-posed optimal Legendre spectral element approximation of the Stokes semi-periodic problem, ICASE Report (1987), no. 87-48.
  • [42] Y. Maday and A. Patera, Spectral element methods for the incompressible Navier-Stokes equations, IN: State-of-the-art surveys on computational mechanics (A90-47176 21-64). New York, American Society of Mechanical Engineers, 1989, p. 71-143. Research supported by DARPA., vol. 1, 1989, pp. 71-143.
  • [43] J. M. Melenk and T. Wurzer, On the stability of the boundary trace of the polynomial $ L^2$-projection on triangles and tetrahedra, Comput. Math. Appl. 67 (2014), no. 4, 944-965. MR 3163888,
  • [44] B. Mercier, J. Osborn, J. Rappaz, and P.-A. Raviart, Eigenvalue approximation by mixed and hybrid methods, Math. Comp. 36 (1981), no. 154, 427-453. MR 606505,
  • [45] Richard Pasquetti and Francesca Rapetti, Spectral element methods on triangles and quadrilaterals: comparisons and applications, Journal of Computational Physics 198 (2004), 349-362.
  • [46] Anthony T Patera, A spectral element method for fluid dynamics: laminar flow in a channel expansion, Journal of computational Physics 54 (1984), no. 3, 468-488.
  • [47] Michael Daniel Samson, Huiyuan Li, and Li-Lian Wang, A new triangular spectral element method I: implementation and analysis on a triangle, Numer. Algorithms 64 (2013), no. 3, 519-547. MR 3120004,
  • [48] Ch. Schwab, $ p$- and $ hp$-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1998. MR 1695813
  • [49] C. Schwab and M. Suri, Mixed $ hp$ finite element methods for Stokes and non-Newtonian flow, Comput. Methods Appl. Mech. Engrg. 175 (1999), no. 3-4, 217-241. MR 1702217,
  • [50] Jie Shen, On fast direct Poisson solver, $ {\rm inf}$-$ {\rm sup}$ constant and iterative Stokes solver by Legendre-Galerkin method, J. Comput. Phys. 116 (1995), no. 1, 184-188. MR 1315216,
  • [51] Jie Shen, Li-Lian Wang, and Huiyuan Li, A triangular spectral element method using fully tensorial rational basis functions, SIAM J. Numer. Anal. 47 (2009), no. 3, 1619-1650. MR 2505867,
  • [52] Spencer J. Sherwin and George Em. Karniadakis, A new triangular and tetrahedral basis for high-order $ (hp)$ finite element methods, Internat. J. Numer. Methods Engrg. 38 (1995), no. 22, 3775-3802. MR 1362003,
  • [53] Rolf Stenberg and Manil Suri, Mixed $ hp$ finite element methods for problems in elasticity and Stokes flow, Numer. Math. 72 (1996), no. 3, 367-389. MR 1367655,
  • [54] Gabor Szegö, Orthogonal polynomials, vol. 23, American Mathematical Society, 1939.
  • [55] T. C. Warburton, S. J. Sherwin, and G. E. Karniadakis, Basis functions for triangular and quadrilateral high-order elements, SIAM J. Sci. Comput. 20 (1999), no. 5, 1671-1695. MR 1694678,
  • [56] J. A. C. Weideman and L. N. Trefethen, The eigenvalues of second-order spectral differentiation matrices, SIAM J. Numer. Anal. 25 (1988), no. 6, 1279-1298. MR 972454,
  • [57] Christian Wieners, A numerical existence proof of nodal lines for the first eigenfunction of the plate equation, Arch. Math. (Basel) 66 (1996), no. 5, 420-427. MR 1383907,
  • [58] C. Wieners, Bounds for the $ N$ lowest eigenvalues of fourth-order boundary value problems, Computing 59 (1997), no. 1, 29-41. MR 1465309,
  • [59] YiDu Yang and Wei Jiang, Upper spectral bounds and a posteriori error analysis of several mixed finite element approximations for the Stokes eigenvalue problem, Sci. China Math. 56 (2013), no. 6, 1313-1330. MR 3063973,
  • [60] Zhimin Zhang, How many numerical eigenvalues can we trust?, J. Sci. Comput. 65 (2015), no. 2, 455-466. MR 3411273,

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2010): 65N35, 65N25, 35P15, 35Q30

Retrieve articles in all journals with MSC (2010): 65N35, 65N25, 35P15, 35Q30

Additional Information

Weikun Shan
Affiliation: Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China – and – University of Chinese Academy of Sciences, Beijing 100190, People’s Republic of China

Huiyuan Li
Affiliation: State Key Laboratory of Computer Science/Laboratory of Parallel Computing, Institute of Software, Chinese Academy of Sciences, Beijing 100190, People’s Republic of China

Keywords: Stokes eigenvalues, triangular spectral element method, generalized orthogonal Koornwinder polynomials, error analysis
Received by editor(s): April 12, 2015
Received by editor(s) in revised form: April 14, 2015, and February 4, 2016
Published electronically: March 29, 2017
Additional Notes: This work was supported by National Natural Science Foundation of China (No. 91130014, 11471312 and 91430216).
Article copyright: © Copyright 2017 American Mathematical Society

American Mathematical Society