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The Superconvergence Phenomenon and Proof of the MAC Scheme for the Stokes Equations on Non-uniform Rectangular Meshes

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Abstract

For decades, the widely used finite difference method on staggered grids, also known as the marker and cell (MAC) method, has been one of the simplest and most effective numerical schemes for solving the Stokes equations and Navier–Stokes equations. Its superconvergence on uniform meshes has been observed by Nicolaides (SIAM J Numer Anal 29(6):1579–1591, 1992), but the rigorous proof is never given. Its behavior on non-uniform grids is not well studied, since most publications only consider uniform grids. In this work, we develop the MAC scheme on non-uniform rectangular meshes, and for the first time we theoretically prove that the superconvergence phenomenon (i.e., second order convergence in the \(L^2\) norm for both velocity and pressure) holds true for the MAC method on non-uniform rectangular meshes. With a careful and accurate analysis of various sources of errors, we observe that even though the local truncation errors are only first order in terms of mesh size, the global errors after summation are second order due to the amazing cancellation of local errors. This observation leads to the elegant superconvergence analysis even with non-uniform meshes. Numerical results are given to verify our theoretical analysis.

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References

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

    Book  MATH  Google Scholar 

  2. Cockburn, B., Kanschat, G., Schötzau, D.: A locally conservative LDG method for the incompressible Navier–Stokes equations. Math. Comp. 74, 1067–1095 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  3. Daly, B.J., Harlow, F.H., Shannon, J.P., Welch, J.E.: The MAC Method, Tech. report LA-3425, Los Alamos Scientific Laboratory, University of California (1965)

  4. Girault, V., Lopez, H.: Finite-element error estimates for the MAC scheme. IMA J. Numer. Anal. 16, 347–379 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gustafsson, B., Nilsson, J.: Boundary conditions and estimates for the steady Stokes equations on staggered grids. J. Sci. Comput. 15, 29–59 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Han, H., Wu, X.: A new mixed finite element formulation and the MAC method for the Stokes equations. SIAM J. Numer. Anal. 35(2), 560–571 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  7. Huang, Y., Li, J., Wu, C., Yang, W.: Superconvergence analysis for linear tetrahedral edge elements. J. Sci. Comput. doi:10.1007/s10915-014-9848-7

  8. Huang, Y., Li, J., Yang, W., Sun, S.: Superconvergence of mixed finite element approximations to 3-D Maxwell’s equations in metamaterials. J. Comp. Phys. 230, 8275–8289 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Huang, Y., Zhang, S.: Supercloseness of the divergence-free finite element solutions on rectangular grids. Commun. Math. Stat. 1, 143–162 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  10. Ito, K., Qiao, Z.: A high order compact MAC finite difference scheme for the Stokes equations: augmented variable approach. J. Comp. Phys. 227, 8177–8190 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kanschat, G.: Divergence-free discontinuous Galerkin schemes for the Stokes equations and the MAC scheme. Int. J. Numer. Meth. Fluids 56, 941–950 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lebedev, V.L.: Difference analogues of orthogonal decompositions, fundamental differential operators and certain boundary-value problems of mathematical physics. Z. Vycisl. Mat. i Mat. Fiz. 4, 449–465 (1964)

    MathSciNet  Google Scholar 

  13. Li, J., Huang, Y.: Time-Domain Finite Element Methods for Maxwell’s Equations in Metamaterials, Springer Series in Computational Mathematics, vol. 43. Springer, Berlin (2013)

    Book  Google Scholar 

  14. Li, M., Tang, T., Fornberg, B.: A compact fourth-order finite difference scheme for the steady incompressible Navier–Stokes equations. Int. J. Numer. Meth. Fluids 20, 1137–1151 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. Minev, P.D.: Remarks on the links between low-order DG methods and some finite-difference schemes for the Stokes problem. Int. J. Numer. Meth. Fluids 58, 307–317 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  16. Monk, P., Süli, E.: A convergence analysis of Yee’s scheme on nonuniform grids. SIAM J. Numer. Anal. 32(2), 393–412 (1994)

    Article  Google Scholar 

  17. Nicolaides, R.A.: Analysis and convergence of the MAC scheme I. The linear problem. SIAM J. Numer. Anal. 29(6), 1579–1591 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  18. Nicolaides, R.A., Wu, X.: Analysis and convergence of the MAC scheme. II. Navier–Stokes equations. Math. Comput. 65, 29–44 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. Qiao, Z., Yao, C., Jia, S.: Superconvergence and extrapolation analysis of a nonconforming mixed finite element approximation for time-harmonic Maxwell’s equations. J. Sci. Comput. 46, 1–19 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rannacher, R., Turek, S.: Simple nonconforming quadrilateral Stokes element. Numer. Methods Partial Differ. Equ. 8, 97–111 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  21. Samelson, R., Temam, R., Wang, C., Wang, S.: Surface pressure Poisson equation formulation of the primitive equations: numerical schemes. SIAM J. Numer. Anal. 41, 1163–1194 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  22. Samelson, R., Temam, R., Wang, C., Wang, S.: A fourth-order numerical method for the planetary geostrophic equations with inviscid geostrophic balance. Numer. Math. 107, 669–705 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  23. Süli, E.: Convergence of finite volume schemes for Poisson’s equation on nonuniform meshes. SIAM J. Numer. Anal. 28(5), 1419–1430 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. Weiser, A., Wheeler, M.F.: On convergence of block-centered finite differences for elliptic problems. SIAM J. Numer. Anal. 25(2), 351–375 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  25. Yang, Y., Shu, C.-W.: Analysis of optimal superconvergence of discontinuous Galerkin method for linear hyperbolic equations. SIAM J. Numer. Anal. 50, 3110–3133 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  26. Zhang, Z. (ed.): Special issues of “Superconvergence and a Posteriori Error Estimates in Finite Element Methods”. Int. J. Numer. Anal. Model 2(1), 1–126 (2005) and 3(3), 255–376 (2006)

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Acknowledgments

J. Li would like to thank UNLV for granting his sabbatical leave during spring 2014 so that he could enjoy his time working on this during his stay at KAUST. S. Sun would like to acknowledge that research reported in this publication was supported in part by KAUST. Finally, the authors like to thank two anonymous referees for their insightful comments that improved this paper.

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Authors and Affiliations

  1. Department of Mathematical Sciences, University of Nevada Las Vegas, Las Vegas, NV, 89154-4020, USA

    Jichun Li

  2. Computational Transport Phenomena Laboratory (CTPL), Physical Sciences and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal, 23955-6900, Kingdom of Saudi Arabia

    Shuyu Sun

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  1. Jichun Li
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  2. Shuyu Sun
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Correspondence to Jichun Li.

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Li, J., Sun, S. The Superconvergence Phenomenon and Proof of the MAC Scheme for the Stokes Equations on Non-uniform Rectangular Meshes. J Sci Comput 65, 341–362 (2015). https://doi.org/10.1007/s10915-014-9963-5

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  • DOI: https://doi.org/10.1007/s10915-014-9963-5

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