Abstract
The Ladyženskaja–Babuška–Brezzi (LBB) condition is a necessary condition for the well-posedness of discrete saddle point problems stemming from discretizing the Stokes equations. In this paper, we prove the LBB condition and provide the (optimal) lower bound for this condition for the triangular spectral method proposed by L. Chen, J. Shen, and C. Xu in [3]. Then this lower bound is used to derive an error estimate for the pressure. Some numerical examples are provided to confirm the theoretical estimates.
Funding source: National Natural Science Foundation of China
Award Identifier / Grant number: 11201077
Award Identifier / Grant number: 11301438
Award Identifier / Grant number: 11471274
Award Identifier / Grant number: 11421110001
Award Identifier / Grant number: 91130002
Funding source: Natural Science Foundation of Fujian Province
Award Identifier / Grant number: JA14034
Award Identifier / Grant number: 2013J05003
Funding statement: The first and third authors were supported by the National Science Foundation of China through grant 11201077, and the NSF of Fujian Province through grants JA14034 and 2013J05003. The second author was supported by the National Science Foundation of China through grant 11301438 and the Postdoctoral Science Foundation of China through grant 2015M580038. Moreover, the research of all authors was supported by the National Science Foundation of China through grants 11471274, 11421110001, and 91130002, and the Sino-German Science Center on the occasion of the Chinese-German Workshop on Computational and Applied Mathematics in Augsburg 2015.
References
[1] Bernardi C., Canuto C. and Maday Y., Generalized inf-sup conditions for Chebyshev spectral approximation of the Stokes problem, SIAM J. Numer. Anal. 25 (1988), no. 6, 1237–1271. 10.1137/0725070Suche in Google Scholar
[2] Braess D. and Schwab C., Approximation on simplices with respect to weighted Sobolev norms, J. Approx. Theory 103 (2000), no. 2, 329–337. 10.1006/jath.1999.3429Suche in Google Scholar
[3] Chen L. Z., Shen J. and Xu C. J., A triangular spectral method for the Stokes equations, Numer. Math. Theor. Mech. Appl. 4 (2011), no. 2, 158–179. 10.4208/nmtma.2011.42s.3Suche in Google Scholar
[4] Chen L. Z., Shen J. and Xu C. J., A unstructured nodal spectral-element method for the Navier–Stokes equations, Commun. Comput. Phys. 12 (2012), no. 1, 315–336. 10.4208/cicp.070111.140711aSuche in Google Scholar
[5] Dubiner M., Spectral methods on triangles and other domains, J. Sci. Comput. 6 (1991), no. 4, 345–390. 10.1007/BF01060030Suche in Google Scholar
[6] Girault V. and Raviart P.-A., Finite Element Methods for Navier–Stokes Equations, Theory and Algorithms, Springer, Berlin, 1986. 10.1007/978-3-642-61623-5Suche in Google Scholar
[7] Heinrichs W. and Loch B. I., Spectral schemes on triangular elements, J. Comput. Phys. 173 (2001), no. 1, 279–301. 10.1006/jcph.2001.6876Suche in Google Scholar
[8] Hesthaven J. S., From electrostatics to almost optimal nodal sets for polynomial interpolation in a simplex, SIAM J. Numer. Anal. 35 (1998), no. 2, 655–676. 10.1137/S003614299630587XSuche in Google Scholar
[9] John J. P., Chebyshev and Fourier Spectral Methods, 2nd ed., Dover Publications, Mineola, 2001. Suche in Google Scholar
[10] Joseph P., Sur une famille de polynomes á deux variables orthogonaux dans un triangle, C. R. Acad. Sci. Paris 245 (1957), no. 26, 2459–2461. Suche in Google Scholar
[11] Karniadakis G. E. and Sherwin S. J., Spectral/hp Element Methods for Computational Fluid Dynamics, 2nd ed., Numer. Math. Sci. Comput., Oxford University Press, New York, 2005. 10.1093/acprof:oso/9780198528692.001.0001Suche in Google Scholar
[12] Koornwinder T., Two-variable analogues of the classical orthogonal polynomials, Theory and Applications of Special Functions, Academic Press, London (1975), 435–495. 10.1016/B978-0-12-064850-4.50015-XSuche in Google Scholar
[13] Li Y. Y., Wang L. L., Li H. Y. and Ma H. P., A new spectral method on triangles, Spectral and High Order Methods for Partial Differential Equations, Lect. Notes Comput. Sci. Eng. 76, Springer, Berlin (2011), 237–246. 10.1007/978-3-642-15337-2_21Suche in Google Scholar
[14] Maday Y., Meiron D., Patera A. and Rønquist E., Analysis of iterative methods for the steady and unsteady Stokes problem: Application to spectral element discretizations, SIAM J. Sci. Comput. 14 (1993), 310–337. 10.1137/0914020Suche in Google Scholar
[15]
Marcus M., Mizel V. and Pinchover Y.,
On the best constant for Hardy’s inequality in
[16] Owens R. G., Spectral approximations on the triangle, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 454 (1998), no. 1971, 857–872. 10.1098/rspa.1998.0189Suche in Google Scholar
[17] Pasquetti R. and Rapetti F., Spectral element methods on triangles and quadrilaterals: Comparisons and applications, J. Comput. Phys. 198 (2004), no. 1, 349–362. 10.1016/j.jcp.2004.01.010Suche in Google Scholar
[18] Pasquetti R. and Rapetti F., Spectral element methods on unstructured meshes: Comparisons and recent advances, J. Sci. Comput. 27 (2006), 377–387. 10.1007/s10915-005-9048-6Suche in Google Scholar
[19] Quarteroni A. and Valli A., Numerical Approximation of Partial Differential Equations, Springer Ser. Comput. Math. 23, Springer, Berlin, 2008. Suche in Google Scholar
[20] Samson M. D., Li H. Y. and Wang L. L., A new triangular spectral element method I: Implementation and analysis on a triangle, Numer. Algor. 64 (2013), no. 3, 519–547. 10.1007/s11075-012-9677-4Suche in Google Scholar
[21] Schwab C., p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics, Clarendon Press, Oxford, 1998. Suche in Google Scholar
[22] Shen J. and Tang T., Spectral and High-order Methods with Applications, Science Press, Beijing, 2006. Suche in Google Scholar
[23] Shen J., Wang L. L. and Li H. Y., A triangular spectral element method using fully tensorial rational basis functions, SIAM J. Numer. Anal. 47 (2009), no. 3, 1619–1650. 10.1137/070702023Suche in Google Scholar
[24] Sherwin S. J. and Karniadakis G. E., A new triangular and tetrahedral basis for high-order (hp) finite element methods, Internat. J. Numer. Methods Engrg. 38 (1995), no. 22, 3775–3802. 10.1002/nme.1620382204Suche in Google Scholar
[25] Taylor M. A., Wingate B. A. and Vincent R. E., An algorithm for computing Fekete points in the triangle, SIAM J. Numer. Anal. 38 (2000), no. 5, 1707–1720. 10.1137/S0036142998337247Suche in Google Scholar
© 2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Stability and Experimental Comparison of Prototypical Iterative Schemes for Total Variation Regularized Problems
- A Robust Weak Galerkin Finite Element Method for Linear Elasticity with Strong Symmetric Stresses
- Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems
- A Hybridizable Discontinuous Galerkin Method for the Time-Harmonic Maxwell Equations with High Wave Number
- A Note on Multilevel Based Error Estimation
- Characterization of Extreme Points of Multi-Stochastic Tensors
- An Adaptive Finite Element Multi-Mesh Approach for Interacting Deformable Objects in Flow
- Computational Comparison of Surface Metrics for PDE Constrained Shape Optimization
- Two-Level Spline Approximations for Two-Dimensional Navier–Stokes Equations
- On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations
Artikel in diesem Heft
- Frontmatter
- Stability and Experimental Comparison of Prototypical Iterative Schemes for Total Variation Regularized Problems
- A Robust Weak Galerkin Finite Element Method for Linear Elasticity with Strong Symmetric Stresses
- Space-Time Discontinuous Galerkin Discretizations for Linear First-Order Hyperbolic Evolution Systems
- A Hybridizable Discontinuous Galerkin Method for the Time-Harmonic Maxwell Equations with High Wave Number
- A Note on Multilevel Based Error Estimation
- Characterization of Extreme Points of Multi-Stochastic Tensors
- An Adaptive Finite Element Multi-Mesh Approach for Interacting Deformable Objects in Flow
- Computational Comparison of Surface Metrics for PDE Constrained Shape Optimization
- Two-Level Spline Approximations for Two-Dimensional Navier–Stokes Equations
- On the Inf-Sup Constant of a Triangular Spectral Method for the Stokes Equations