Skip to main content
SpringerLink
Log in

An adaptive least-squares spectral collocation method with triangular elements for the incompressible Navier–Stokes equations

  • Original Paper
  • Published:
Journal of Engineering Mathematics Aims and scope Submit manuscript

Abstract

A least-squares spectral collocation scheme for the incompressible Navier–Stokes equations is proposed. Grid refinement is performed by means of adaptive triangular elements. On each triangle the Fekete nodes are employed for the collocation of the differential equation. On the element interfaces continuity of the functions is enforced in the least-squares sense. Equal-order Dubiner polynomials are used to obtain a stable spectral scheme. The collocation conditions and the interface conditions lead to an overdetermined system that can be solved efficiently by least-squares. The solution technique only involves symmetric positive-definite linear systems. The approach is first applied to the Poisson equation and then extended to singular perturbation problems where least-squares have a stabilizing effect. By adaptivity, a suitable decomposition of the domain is found where the boundary layer is well resolved. Finally, the method is successfully applied to the regularized driven-cavity flow problem. Numerical simulations confirm the high accuracy of the proposed spectral least-squares scheme.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Canuto C, Hussaini MY, Quarteroni A, Zang TA (1989) Spectral methods in fluid dynamics, Springer Series in Computational physics. Springer-Verlag, Berlin-Heidelberg-New York

    Google Scholar 

  2. Gottlieb D, Orszag SA (1977) Numerical analysis of spectral methods: theory and applications. CBMS-NSF Regional Conference Series in Applied Mathematics No. 26. SIAM, Philadelphia

    Google Scholar 

  3. Orszag SA (1980) Spectral methods for problems in complex geometries. J Comput Phys 37:70–92

    Article  MATH  MathSciNet  Google Scholar 

  4. Deville MO, Fischer PF, Mund EH (2002) High-order methods for incompressible fluid flow. Cambridge monographs on applied and computational mathematics. Cambridge University Press

  5. Gerritsma MI, Proot MJ (2002) Analysis of a discontinuous least-squares spectral element method. J Sci Comput 17:297–306

    Article  MATH  MathSciNet  Google Scholar 

  6. Heinrichs W (2003) Least-squares spectral collocation for discontinuous and singular perturbation problems. J Comput Appl Math 157:329–345

    Article  MATH  MathSciNet  Google Scholar 

  7. Eisen H, Heinrichs W (1992) A new method of stabilization for singular perturbation problems with spectral methods. SIAM J Numer Anal 29:107–122

    Article  MATH  MathSciNet  Google Scholar 

  8. Heinrichs W (1992) A stabilized multidomain approach for singular perturbation problems. J Sci Comput 7:95–125

    Article  MATH  MathSciNet  Google Scholar 

  9. Heinrichs W (1994) Spectral viscosity for convection dominated flow. J Sci Comput 9:137–148

    Article  MATH  MathSciNet  Google Scholar 

  10. Henderson RD (1999) Adaptive spectral element methods for turbulence and transition. In: Barth TJ, Deconinck H (eds) High order methods for computational physics. Springer, Berlin, pp 225–324

    Google Scholar 

  11. Verfürth R (1996) A review of a posteriori error estimation and adaptive mesh refinement. Wiley-Teubner, Chichester-New York-Stuttgart

    MATH  Google Scholar 

  12. Solin P (2005) Partial differential equations and the finite element method. Wiley, New York

    Google Scholar 

  13. Heinrichs W (1998) Spectral collocation on triangular elements. J Comput Phys 145:743–757

    Article  MATH  MathSciNet  Google Scholar 

  14. Heinrichs W, Loch B (2001) Spectral schemes on triangular elements. J Comput Phys 173:279–301

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Karniadakis G, Sherwin SJ (1999) Spectral/hp element methods for CFD numerical mathematics and computation. Oxford Univ. Press, London

    Google Scholar 

  16. Mavriplis C, Van Rosendale A (1993) Triangular spectral elements for incompressible fluid flow. ICASE Rep pp 93–100

  17. Bos L (1983) Bounding the Lebesgue function for Lagrange interpolation in a simplex. J Approx Theory 38:43–59

    Article  MATH  MathSciNet  Google Scholar 

  18. Chen Q, Babuška I (1995) Approximate optimal points for polynomial interpolation of real functions in an interval and in a triangle. Comput Meth Appl Mech Engng 128:405–417

    Article  MATH  Google Scholar 

  19. Hesthaven JS (1998) From electrostatics to almost optimal nodal sets for polynomial interpolation. SIAM J Numer Anal 35:655–676

    Article  MATH  MathSciNet  Google Scholar 

  20. Taylor MA, Wingate BA, Vincent RE (2000) An algorithm for computing Fekete points in the triangle. SIAM J Numer Anal 38:1707–1720

    Article  MATH  MathSciNet  Google Scholar 

  21. Pasquetti R, Rapetti F (2004) Spectral element methods on triangles and quadrilaterals: comparisons and applications. J Comput Phys 198:349–362

    Article  MATH  ADS  Google Scholar 

  22. Dubiner M (1993) Spectral methods on triangles and other domains. J Sci Comput 6:345–390

    Article  MathSciNet  Google Scholar 

  23. Davis TA (2005) UMFPACK – Version 4.4 user guide. http://www.cise.ufl.edu/research/sparse/umfpack/: Dept. of Computer and Information Science and Engineering - Univ. of Florida

  24. Davis TA (2004) UMFPACK – an unsymmetric-pattern multifrontal method with a column pre-ordering strategy. ACM Trans Math Software 30(2):196–199

    Article  MATH  MathSciNet  Google Scholar 

  25. Deang JM, Gunzburger MD (1998) Issues related to least-squares finite element methods for the Stokes equations. SIAM J Sci Comput 20:878–906

    Article  MathSciNet  Google Scholar 

  26. Jiang B-N (1998) On the least-squares method. Comput Meth Appl Mech Engng 152:239–257

    Article  MATH  Google Scholar 

  27. Jiang B-N, Chang CL (1990) Least-squares finite elements for the Stokes problem. Comput Meth Appl Mech Engng 78:297–311

    Article  MATH  MathSciNet  Google Scholar 

  28. Jiang B-N (1992) A least-squares finite element method for incompressible Navier–Stokes problems. Int J Numer Meth Fluids 14:843–859

    Article  MATH  Google Scholar 

  29. Jiang B-N, Povinelli L (1990) Least-squares finite element method for fluid dynamics. Comput Meth Appl Mech Engng 81:13–37

    Article  MATH  MathSciNet  Google Scholar 

  30. Bernardi C, Canuto C, Maday Y (1986) Generalized inf-sup condition for Chebyshev approximations to Navier–Stokes equations. C. R. Acad Sci Paris 303:971–974

    MATH  MathSciNet  Google Scholar 

  31. Bernardi C, Maday Y (1992) Approximations spectrale de problémes aux limites elliptiques. Springer-Verlag, Berlin

    Google Scholar 

  32. Rønquist E (1988) Optimal spectral element methods for the unsteady three dimensional incompressible Navier–Stokes Equations. Ph. D. thesis, Massachusetts Institute of Technology, Cambridge

  33. Heinrichs W (1993) Splitting techniques for the pseudospectral approximation of the unsteady Stokes equations. SIAM J Numer Anal 30(1):19–39

    Article  MATH  MathSciNet  Google Scholar 

  34. Proot MJ, Gerritsma MI (2002) A least-squares spectral element formulation for the Stokes problem. J Sci Comput 17:285–296

    Article  MATH  MathSciNet  Google Scholar 

  35. Proot MJ, Gerritsma MI (2002) Least-squares spectral elements applied to the Stokes problem. J Comput Phys 181: 454–477

    Article  MATH  ADS  MathSciNet  Google Scholar 

  36. Pontaza JP, Reddy JN (2003) Spectral/hp least-squares finite element formulation for the Navier–Stokes equations. J Comput Phys 190:523–549

    Article  MATH  ADS  MathSciNet  Google Scholar 

  37. Pontaza JP, Reddy JN (2004) Space-time coupled spectral/hp least-squares finite element formulation for the incompressible avier–Stokes equations. J Comput Phys 197:418–459

    Article  MATH  ADS  MathSciNet  Google Scholar 

  38. Proot MJ, Gerritsma MI (2005) Application of the least-squares spectral element method using Chebyshev polynomials to solve the incompressible Navier–Stokes equations. Numer Algor 38:155–172

    Article  MATH  ADS  MathSciNet  Google Scholar 

  39. Pontaza JP, Reddy JN (2006) Least-squares finite element formulations for viscous incompressible and compressible fluid flows. Comput Meth Appl Mech Engng 195:2454–2494

    Article  MATH  MathSciNet  Google Scholar 

  40. Heinrichs W (2004) Least-squares spectral collocation for the Navier–Stokes equations. J Sci Comput 21:81–90

    Article  MATH  MathSciNet  Google Scholar 

  41. Gerritsma MI, Phillips TN (1998) Discontinuous spectral element approximation for the velocity-pressure-stress formulation of the Stokes problem. Internat J Numer Meth Engng 43:1401–1419

    Article  MATH  MathSciNet  Google Scholar 

  42. Jiang B-N (1998) The least-squares finite element method, Theory and applications in computational fluid dynamics and electromagnetics. Springer-Verlag, Berlin

    Google Scholar 

  43. Mulholland LS, Huang W-Z, Sloan DM (1998) Pseudospectral solution of near-singular problems using numerical coordinate transformations based on adaptivity. SIAM J Sci Comput 19:1261–1289

    Article  MATH  MathSciNet  Google Scholar 

  44. Haschke H, Heinrichs W (2001) Splitting techniques with staggered grids for the Navier–Stokes equations in the 2D case. J Comput Phys 168:131–154

    Article  MATH  MathSciNet  Google Scholar 

  45. Botella O (1997) On the solution of the Navier–Stokes equations using Chebyshev projection schemes with third-order accuracy in time. Comput Fluids 26:107–116

    Article  MATH  Google Scholar 

  46. Botella O, Peyret R (1998) Benchmark spectral results on the lid-driven cavity flow. Comput Fluids 27:421–433

    Article  MATH  Google Scholar 

  47. Ghia U, Ghia KN, Shin CT (1982) High-Re solutions for incompressible flow using the Navier–Stokes equations and a multigrid method. J Comput Phys 48:387–411

    Article  MATH  ADS  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Duisburg–Essen, Ingenieurmathematik, Universitätsstr. 3, D-45117, Essen, Germany

    Wilhelm Heinrichs

Authors
  1. Wilhelm Heinrichs
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wilhelm Heinrichs.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Heinrichs, W. An adaptive least-squares spectral collocation method with triangular elements for the incompressible Navier–Stokes equations. J Eng Math 56, 337–350 (2006). https://doi.org/10.1007/s10665-006-9081-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10665-006-9081-y

Keywords

Search

Navigation