Stokes elements on cubic meshes yielding divergence-free approximations

Abstract

Conforming piecewise polynomial spaces with respect to cubic meshes are constructed for the Stokes problem in arbitrary dimensions yielding exactly divergence-free velocity approximations. The derivation of the finite element pair is motivated by a smooth de Rham complex that is well-suited for the Stokes problem. We derive the stability and convergence properties of the new elements as well as the construction of reduced elements with less global unknowns.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2

References

  1. 1.

    Arnold, D.N., Boffi, D., Bonizzoni, F.: Finite element differential forms on curvilinear cubic meshes and their approximation properties. Numer. Math. 129(1), 1–20 (2015)

  2. 2.

    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. (N.S.) 47(2), 281–354 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Austin, T.M., Manteuffel, T.A., McCormick, S.: A robust multilevel approach for minimizing \(H(div)\)-dominated functionals in an \(H1\)-conforming finite element space. Numer. Linear Algebra Appl. 11(2–3), 115–140 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Boffi, D., Brezzi, F., Demkowicz, L.F., Durán, R.G., Falk, R.S., Fortin, M.: Mixed finite elements, compatibility conditions, and applications, Lectures given at the C.I.M.E. Summer School held in Cetraro. In: Boffi, D., Gastaldi, L. (Eds.) Lecture Notes in Mathematics, 1939. Springer, Berlin, Fondazione C.I.M.E., Florence (2008)

  6. 6.

    Boffi, D., Brezzi, F., Fortin, M.: Mixed Finite Element Methods and Applications, Springer Series in Computational Mathematics, 44. Springer, Heidelberg (2013)

    Google Scholar 

  7. 7.

    Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Springer, Berlin (2008)

    Google Scholar 

  8. 8.

    Buffa, A., de Falco, C., Sangalli, G.: IsoGeometric analysis: stable elements for the 2D Stokes equation. Int. J. Numer. Methods Fluids 65(11–12), 1407–1422 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Buffa, A., Rivas, J., Sangalli, G., Vázquez, R.: Isogeometric discrete differential forms in three dimensions. SIAM J. Numer. Anal. 49(2), 818–844 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  10. 10.

    Evans, J.A.: Divergence-free B-spline discretizations for viscous flows. Ph.D. Thesis, The University of Texas, Austin, Tx (2011)

  11. 11.

    Evans, J.A., Hughes, T.J.R.: Isogeometric divergence-conforming B-splines for the steady Navier-Stokes equations. Math. Models Methods Appl. Sci. 23(8), 1421–1478 (2013a)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Evans, J.A., Hughes, T.J.R.: Isogeometric divergence-conforming B-splines for the Darcy-Stokes-Brinkman equations. Math. Models Methods Appl. Sci. 23(4), 671–741 (2013b)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Falk, R.S., Neilan, M.: Stokes complexes and the construction of stable finite element methods with pointwise mass conservation. SIAM J. Numer. Anal. 51(2), 1308–1326 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  14. 14.

    Galvin, K.J., Linke, A., Rebholz, L.G., Wilson, N.E.: Stabilizing poor mass conservation in incompressible flow problems with large irrotational forcing and application to thermal convection. Comput. Methods Appl. Mech. Eng. 237–240, 166–176 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Girault, V., Raviart, P.-A.: Finite Element Methods for the Navier-Stokes Equations. Springer, Berlin (1986)

    Google Scholar 

  16. 16.

    Guzmán, J., Neilan, M.: Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comp. 83(285), 15–36 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Huang, Y., Zhang, S.: A lowest order divergence-free finite element on rectangular grids. Front. Math. China 6(2), 253–270 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Linke, A.: Collision in a cross-shaped domain—a steady 2d Navier-Stokes example demonstrating the importance of mass conservation in CFD. Comput. Methods Appl. Mech. Eng. 198(41–44), 3278–3286 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Monk, P.: Finite Element Methods for Maxwell’s Equations, Numerical Mathematics and Scientific Computation. Oxford University Press, New York (2003)

    Google Scholar 

  20. 20.

    Nédélec, J.-C.: Mixed finite elements in \(R^3\). Numer. Math. 35(3), 315–341 (1980)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Neilan, M.: Discrete and conforming smooth de Rham complexes in three dimensions, Math. Comp., to appear

  22. 22.

    Nitsche, J.: Über ein Variationsprinzip zur Lösung von Dirichlet-Problemen bei Verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind. Abh. Math. Sem. Univ. Hamburg 36, 9–15 (1971)

    MathSciNet  Article  MATH  Google Scholar 

  23. 23.

    Scott, L.R., Vogelius, M.: Norm estimates for a maximal right inverse of the divergence operator in spaces of piecewise polynomials. RAIRO Model. Math. Anal. Numer. 19(1), 111–143 (1985)

    MathSciNet  MATH  Google Scholar 

  24. 24.

    Scott, R.L., Zhang, S.: Finite element interpolation of non smooth functions satisfying boundary conditions. Math. Comp. 54(190), 483–493 (1990)

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Tai, X.-C., Winther, R.: A discrete de Rham complex with enhanced smoothness. Calcolo 43, 287–306 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  26. 26.

    Zhang, S.: A family of \(Q_{k+1, k}\times Q_{k, k+1}\) divergence-free finite elements on rectangular grids. SIAM J. Numer. Anal. 47(3), 2090–2107 (2009)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgments

This work is supported in part by the National Science Foundation grant DMS-1417980 and the Alfred Sloan Foundation.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Michael Neilan.

Appendix: A calculus identity

Appendix: A calculus identity

Lemma 8

Let \(\{\varvec{a}_i\}_{i=1}^n \subset {\mathbb {R}}^n\) be a set of constant orthonormal (column) vectors.

Then there holds

$$\begin{aligned} {\mathrm{div}}\,{\varvec{v}}= \sum _{i=1}^n \frac{{\partial }{\varvec{v}}}{{\partial }\varvec{a}_i} \cdot \varvec{a}_i. \end{aligned}$$

Proof

Let A be the orthonormal matrix \(A = [\varvec{a}_1| \varvec{a}_2|\cdots |\varvec{a}_n]\in {\mathbb {R}}^{n\times n}\), and define \(\hat{{\varvec{v}}}(\hat{x}) = A^{-1} {\varvec{v}}(x) = A^T {\varvec{v}}(x)\), where \(x = A \hat{x}\). We then have \(D {\varvec{v}}(x) = A \hat{D} \hat{{\varvec{v}}}(\hat{x}) A^{T}\) by the chain rule [5].

Therefore, since the trace is invariant under similarity transforms, and since A is orthonormal, we have

$$\begin{aligned} \sum _{i=1}^n \frac{{\partial }{\varvec{v}}}{{\partial }\varvec{a}_i} \cdot \varvec{a}_i&= \sum _{i=1}^n \varvec{a}^T_i (D{\varvec{v}})\varvec{a}_i = \sum _{i=1}^n (\varvec{a}^T_i A) \hat{D} \hat{{\varvec{v}}} (\varvec{a}^T_i A)^T =\sum _{i=1}^n \frac{{\partial }\hat{v}^{(i)}}{{\partial }\hat{x}_i}\\&= \widehat{{\mathrm{div}}\,}\hat{{\varvec{v}}}= \mathrm{tr}(\hat{D}\hat{{\varvec{v}}}) = \mathrm{tr}(D{\varvec{v}}) = {\mathrm{div}}\,{\varvec{v}}. \end{aligned}$$

\(\square \)

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Neilan, M., Sap, D. Stokes elements on cubic meshes yielding divergence-free approximations. Calcolo 53, 263–283 (2016). https://doi.org/10.1007/s10092-015-0148-x

Download citation

Keywords

  • Stokes
  • Finite element analysis
  • Divergence-free

Mathematics Subject Classification

  • 65N30
  • 65N12