A Comparison of HDG Methods for Stokes Flow (original) (raw)

Abstract

In this paper, we compare hybridizable discontinuous Galerkin (HDG) methods for numerically solving the velocity-pressure-gradient, velocity-pressure-stress, and velocity-pressure-vorticity formulations of Stokes flow. Although they are defined by using different formulations of the Stokes equations, the methods share several common features. First, they use polynomials of degree k for all the components of the approximate solution. Second, they have the same globally coupled variables, namely, the approximate trace of the velocity on the faces and the mean of the pressure on the elements. Third, they give rise to a matrix system of the same size, sparsity structure and similar condition number. As a result, they have the same computational complexity and storage requirement. And fourth, they can provide, by means of an element-by element postprocessing, a new approximation of the velocity which, unlike the original velocity, is divergence-free and H(div)-conforming. We present numerical results showing that each of the approximations provided by these three methods converge with the optimal order of k+1 in L 2 for any _k_≥0. We also display experiments indicating that the postprocessed velocity is a better approximation than the original approximate velocity. It converges with an additional order than the original velocity for the gradient-based HDG, and with the same order for the vorticity-based HDG methods. For the stress-based HDG methods, it seems to converge with an additional order for even polynomial degree approximations. Finally, the numerical results indicate that the method based on the velocity-pressure-gradient formulation provides the best approximations for similar computational complexity.

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References

  1. Baker, G.A., Jureidini, W.N., Karakashian, O.A.: Piecewise solenoidal vector fields and the Stokes problem. SIAM J. Numer. Anal. 27, 1466–1485 (1990)
    Article MATH MathSciNet Google Scholar
  2. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods. Springer, Berlin (1991)
    MATH Google Scholar
  3. Carrero, J., Cockburn, B., Schötzau, D.: Hybridized globally divergence-free LDG methods, I: the Stokes problem. Math. Comput. 75, 533–563 (2006)
    MATH Google Scholar
  4. Cockburn, B., Gopalakrishnan, J.: Incompressible finite elements via hybridization, part I: the stokes system in two space dimensions. SIAM J. Numer. Anal. 43(4), 1627–1650 (2005)
    Article MATH MathSciNet Google Scholar
  5. Cockburn, B., Gopalakrishnan, J.: Incompressible finite elements via hybridization, part II: the Stokes system in three space dimensions. SIAM J. Numer. Anal. 43(4), 1651–1672 (2005)
    Article MATH MathSciNet Google Scholar
  6. Cockburn, B., Gopalakrishnan, J.: The derivation of hybridizable discontinuous Galerkin methods for Stokes flow. SIAM J. Numer. Anal. 47, 1092–1125 (2009)
    Article MATH MathSciNet Google Scholar
  7. Cockburn, B., Gopalakrishnan, J., Guzmán, J.: A new elasticity element made for enforcing weak stress symmetry. Math. Comput. (to appear)
  8. Cockburn, B., Gopalakrishnan, J., Nguyen, N.C., Peraire, J., Sayas, F.-J.: Analysis of an HDG method for Stokes flow. Math. Comput. (to appear)
  9. Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier-Stokes equations. Math. Comput. 74, 1067–1095 (2005)
    MATH Google Scholar
  10. Cockburn, B., Kanschat, G., Schötzau, D.: A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 31, 61–73 (2007)
    Article MATH MathSciNet Google Scholar
  11. Cockburn, B., Kanschat, G., Schötzau, D., Schwab, C.: Local discontinuous Galerkin methods for the Stokes system. SIAM J. Numer. Anal. 40(1), 319–343 (2002)
    Article MATH MathSciNet Google Scholar
  12. Fortin, M., Glowinski, R.: Augmented Lagrangian Methods. Studies in Mathematics and its Applications, vol. 15. North-Holland, Amsterdam (1983). Applications to the numerical solution of boundary value problems, Translated from the French by B. Hunt and D.C. Spicer
    Google Scholar
  13. Kovasznay, L.I.G.: Laminar flow behind two-dimensional grid. Proc. Camb. Philos. Soc. 44, 58–62 (1948)
    Article MATH MathSciNet Google Scholar
  14. Montlaur, A., Fernández-Méndez, S., Huerta, A.: Discontinuous Galerkin methods for the Stokes equations using divergence-free approximations. Int. J. Numer. Methods Fluids 57, 1071–1092 (2008)
    Article MATH Google Scholar
  15. Nédélec, J.-C.: A new family of mixed finite elements in R 3. Numer. Math. 50, 57–81 (1986)
    Article MATH MathSciNet Google Scholar
  16. Nguyen, N.C., Peraire, J., Cockburn, B.: A hybridizable discontinuous Galerkin method for Stokes flow. Comput. Methods Appl. Mech. Eng. 199, 582–597 (2010)
    Article Google Scholar
  17. Soon, S.-C., Cockburn, B., Stolarski, H.: A hybridizable discontinuous Galerkin method for linear elasticity. Int. J. Numer. Methods Eng. 80(8), 1058–1092 (2009)
    Article MATH MathSciNet Google Scholar
  18. Verfürth, R.: A posteriori error estimators for the Stokes equations. Numer. Math. 55, 309–325 (1989)
    Article MATH MathSciNet Google Scholar
  19. Wang, J., Ye, X.: New Finite element methods in computational fluid dynamics by H(div) elements. SIAM J. Numer. Anal. 45, 1269–1286 (2007)
    Article MATH MathSciNet Google Scholar

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Authors and Affiliations

  1. School of Mathematics, University of Minnesota, Minneapolis, MN, 55455, USA
    B. Cockburn
  2. Department of Aeronautics and Astronautics, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA
    N. C. Nguyen & J. Peraire

Authors

  1. B. Cockburn
  2. N. C. Nguyen
  3. J. Peraire

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Correspondence toN. C. Nguyen.

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Cockburn, B., Nguyen, N.C. & Peraire, J. A Comparison of HDG Methods for Stokes Flow.J Sci Comput 45, 215–237 (2010). https://doi.org/10.1007/s10915-010-9359-0

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