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Superconvergent Pseudostress-Velocity Finite Element Methods for the Oseen Equations

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Abstract

We present a priori and superconvergence error estimates of mixed finite element methods for the pseudostress-velocity formulation of the Oseen equation. In particular, we derive superconvergence estimates for the velocity and a priori error estimates under unstructured grids, and obtain superconvergence results for the pseudostress under certain structured grids. A variety of numerical experiments validate the theoretical results and illustrate the effectiveness of the superconvergent recovery-based adaptive mesh refinement. It is also numerically shown that the proposed postprocessing yields apparent superconvergence in a benchmark problem for the incompressible Navier–Stokes equation.

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Data Availability

Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.

Code Availability Statement

The code used in this study is available from the authors upon request.

References

  1. Acosta, G., Apel, T., Durán, R.G., Lombardi, A.L.: Error estimates for Raviart-Thomas interpolation of any order on anisotropic tetrahedra. Math. Comp. 80(273), 141–163 (2011). https://doi.org/10.1090/S0025-5718-2010-02406-8

    Article  MathSciNet  MATH  Google Scholar 

  2. Antonietti, P.F., Beirão da Veiga, L., Mora, D., Verani, M.: A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52(1), 386–404 (2014). https://doi.org/10.1137/13091141X

    Article  MathSciNet  MATH  Google Scholar 

  3. Apel, T., Kempf, V.: Brezzi-Douglas-Marini interpolation of any order on anisotropic triangles and tetrahedra. SIAM J. Numer. Anal. 58(3), 1696–1718 (2020). https://doi.org/10.1137/19M1302910

    Article  MathSciNet  MATH  Google Scholar 

  4. Arnold, D.N., Douglas, J., Jr., Gupta, C.P.: A family of higher order mixed finite element methods for plane elasticity. Numer. Math. 45(1), 1–22 (1984). https://doi.org/10.1007/BF01379659

    Article  MathSciNet  MATH  Google Scholar 

  5. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15, 1–155 (2006). https://doi.org/10.1017/S0962492906210018

    Article  MathSciNet  MATH  Google Scholar 

  6. Arnold, D.N., Falk, R.S., Winther, R.: Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47(2), 281–354 (2010). https://doi.org/10.1090/S0273-0979-10-01278-4

    Article  MathSciNet  MATH  Google Scholar 

  7. Arnold, D.N., Li, L.: Finite element exterior calculus with lower-order terms. Math. Comp. 86(307), 2193–2212 (2017). https://doi.org/10.1090/mcom/3158

    Article  MathSciNet  MATH  Google Scholar 

  8. Babuška, I., Rheinboldt, W.C.: Error estimates for adaptive finite element computations. SIAM J. Numer. Anal. 15(4), 736–754 (1978). https://doi.org/10.1137/0715049

    Article  MathSciNet  MATH  Google Scholar 

  9. Bank, R.E., Li, Y.: Superconvergent recovery of Raviart-Thomas mixed finite elements on triangular grids. J. Sci. Comput. 81(3), 1882–1905 (2019). https://doi.org/10.1007/s10915-019-01068-0

    Article  MathSciNet  MATH  Google Scholar 

  10. Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators: I: grids with superconvergence. SIAM J. Numer. Anal. 41(6), 2294–2312 (2003). https://doi.org/10.1137/S003614290139874X

    Article  MathSciNet  MATH  Google Scholar 

  11. Barrios, T.P., Cascón, J.M., González, M.: Augmented mixed finite element method for the Oseen problem: a priori and a posteriori error analyses. Comput. Methods Appl. Mech. Engrg. 313, 216–238 (2017). https://doi.org/10.1016/j.cma.2016.09.012

    Article  MathSciNet  MATH  Google Scholar 

  12. Barrios, T.P., Cascón, J.M., González, M.: On an adaptive stabilized mixed finite element method for the Oseen problem with mixed boundary conditions. Comput. Methods Appl. Mech. Eng. (2020). https://doi.org/10.1016/j.cma.2020.113007

    Article  MathSciNet  MATH  Google Scholar 

  13. Behr, M.A., Franca, L.P., Tezduyar, T.E.: Stabilized finite element methods for the velocity-pressure-stress formulation of incompressible flows. Comput. Methods Appl. Mech. Eng. 104(1), 31–48 (1993). https://doi.org/10.1016/0045-7825(93)90205-C

    Article  MathSciNet  MATH  Google Scholar 

  14. Boffi, D., Brezzi, F., Fortin, M.: Mixed finite element methods and applications. In: Springer Series in Computational Mathematics. Springer, Heidelberg (2013)

    Google Scholar 

  15. Bramble, J.H., Xu, J.: A local post-processing technique for improving the accuracy in mixed finite-element approximations. SIAM J. Numer. Anal. 26(6), 1267–1275 (1989). https://doi.org/10.1137/0726073

    Article  MathSciNet  MATH  Google Scholar 

  16. Brandts, J.H.: Superconvergence and a posteriori error estimation for triangular mixed finite elements. Numer. Math. 68(3), 311–324 (1994). https://doi.org/10.1007/s002110050064

    Article  MathSciNet  MATH  Google Scholar 

  17. Brandts, J.H.: Superconvergence for triangular order \(k=1\) Raviart-Thomas mixed finite elements and for triangular standard quadratic finite element methods. Appl. Numer. Math. 34(1), 39–58 (2000). https://doi.org/10.1016/S0168-9274(99)00034-3

    Article  MathSciNet  MATH  Google Scholar 

  18. Brezzi, F., Douglas, J., Jr., Durán, R., Fortin, M.: Mixed finite elements for second order elliptic problems in three variables. Numer. Math. 51(2), 237–250 (1987). https://doi.org/10.1007/BF01396752

    Article  MathSciNet  MATH  Google Scholar 

  19. Brezzi, F., Douglas, J., Jr., Fortin, M., Marini, L.D.: Efficient rectangular mixed finite elements in two and three space variables. RAIRO Modél. Math. Anal. Numér. 21(4), 581–604 (1987). https://doi.org/10.1051/m2an/1987210405811

    Article  MathSciNet  MATH  Google Scholar 

  20. Brezzi, F., Douglas, J., Jr., Marini, L.D.: Two families of mixed finite elements for second order elliptic problems. Numer. Math. 2(47), 217–235 (1985)

    Article  MathSciNet  Google Scholar 

  21. Cáceres, E., Gatica, G.N.: A mixed virtual element method for the pseudostress-velocity formulation of the Stokes problem. IMA J. Numer. Anal. 37(1), 296–331 (2017). https://doi.org/10.1093/imanum/drw002

    Article  MathSciNet  MATH  Google Scholar 

  22. Cáceres, E., Gatica, G.N., Sequeira, F.A.: A mixed virtual element method for the Brinkman problem. Math. Models Methods Appl. Sci. 27(4), 707–743 (2017). https://doi.org/10.1142/S0218202517500142

    Article  MathSciNet  MATH  Google Scholar 

  23. Cai, Z., Tong, C., Vassilevski, P.S., Wang, C.: Mixed finite element methods for incompressible flow: stationary Stokes equations. Numer. Methods Partial Differ. Equ. 26(4), 957–978 (2010). https://doi.org/10.1002/num.20467

    Article  MathSciNet  MATH  Google Scholar 

  24. Cai, Z., Wang, C., Zhang, S.: Mixed finite element methods for incompressible flow: stationary Navier-Stokes equations. SIAM J. Numer. Anal. 48(1), 79–94 (2010). https://doi.org/10.1137/080718413

    Article  MathSciNet  MATH  Google Scholar 

  25. Cai, Z., Wang, Y.: A multigrid method for the pseudostress formulation of Stokes problems. SIAM J. Sci. Comput. 29(5), 2078–2095 (2007). https://doi.org/10.1137/060661429

    Article  MathSciNet  MATH  Google Scholar 

  26. Cai, Z., Zhang, S.: Mixed methods for stationary Navier-Stokes equations based on pseudostress-pressure-velocity formulation. Math. Comput. 81(280), 1903–1927 (2012). https://doi.org/10.1090/S0025-5718-2012-02585-3

    Article  MathSciNet  MATH  Google Scholar 

  27. Carrero, J., Cockburn, B., Schötzau, D.: Hybridized globally divergence-free LDG methods: I: the Stokes problem. Math. Comput. 75(254), 533–563 (2006). https://doi.org/10.1090/S0025-5718-05-01804-1

    Article  MathSciNet  MATH  Google Scholar 

  28. Carstensen, C., Gallistl, D., Schedensack, M.: Quasi-optimal adaptive pseudostress approximation of the Stokes equations. SIAM J. Numer. Anal. 51(3), 1715–1734 (2013)

    Article  MathSciNet  Google Scholar 

  29. Carstensen, C., Kim, D., Park, E.J.: A priori and a posteriori pseudostress-velocity mixed finite element error analysis for the Stokes problem. SIAM J. Numer. Anal. 49(6), 2501–2523 (2011)

    Article  MathSciNet  Google Scholar 

  30. Cesmelioglu, A., Cockburn, B., Nguyen, N.C., Peraire, J.: Analysis of HDG methods for Oseen equations. J. Sci. Comput. 55(2), 392–431 (2013). https://doi.org/10.1007/s10915-012-9639-y

    Article  MathSciNet  MATH  Google Scholar 

  31. Cesmelioglu, A., Cockburn, B., Qiu, W.: Analysis of a hybridizable discontinuous Galerkin method for the steady-state incompressible Navier-Stokes equations. Math. Comp. 86(306), 1643–1670 (2017). https://doi.org/10.1090/mcom/3195

    Article  MathSciNet  MATH  Google Scholar 

  32. Chen, X., Li, Y., Drapaca, C., Cimbala, J.: A unified framework of continuous and discontinuous Galerkin methods for solving the incompressible Navier-Stokes equation. J. Comp. Phys. (2020). https://doi.org/10.1016/j.jcp.2020.109799

    Article  MathSciNet  MATH  Google Scholar 

  33. Christiansen, S.H., Winther, R.: Smoothed projections in finite element exterior calculus. Math. Comput. 77(262), 813–829 (2008)

    Article  MathSciNet  Google Scholar 

  34. Cockburn, B., Sayas, F.J.: Divergence-conforming HDG methods for Stokes flows. Math. Comput. 83(288), 1571–1598 (2014). https://doi.org/10.1090/S0025-5718-2014-02802-0

    Article  MathSciNet  MATH  Google Scholar 

  35. Cui, M., Ye, X.: Superconvergence of finite volume methods for the Stokes equations. Numer. Methods Partial Differ. Equ. 25(5), 1212–1230 (2009). https://doi.org/10.1002/num.20399

    Article  MathSciNet  MATH  Google Scholar 

  36. Demlow, A.: Suboptimal and optimal convergence in mixed finite element methods. SIAM J. Numer. Anal. 39(6), 1938–1953 (2002). https://doi.org/10.1137/S0036142900376900

    Article  MathSciNet  MATH  Google Scholar 

  37. Di Pietro, D.A., Ern, A.: Mathematical aspects of discontinuous Galerkin methods, Mathématiques & Applications (Berlin) [Mathematics & Applications], Springer, Heidelberg (2012). https://doi.org/10.1007/978-3-642-22980-0

  38. Dörfler, W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33(3), 1106–1124 (1996). https://doi.org/10.1137/0733054

    Article  MathSciNet  MATH  Google Scholar 

  39. Douglas, J., Jr., Roberts, J.E.: Global estimates for mixed methods for second order elliptic equations. Math. Comput. 44(169), 39–52 (1985). https://doi.org/10.2307/2007791

    Article  MathSciNet  MATH  Google Scholar 

  40. Durán, R.: Superconvergence for rectangular mixed finite elements. Numer. Math. 58(3), 287–298 (1990). https://doi.org/10.1007/BF01385626

    Article  MathSciNet  MATH  Google Scholar 

  41. Egger, H., Schöberl, J.: A hybrid mixed discontinuous Galerkin finite-element method for convection-diffusion problems. IMA J. Numer. Anal. 30(4), 1206–1234 (2010). https://doi.org/10.1093/imanum/drn083

    Article  MathSciNet  MATH  Google Scholar 

  42. Eichel, H., Tobiska, L., Xie, H.: Supercloseness and superconvergence of stabilized low-order finite element discretizations of the Stokes problem. Math. Comput. 80(274), 697–722 (2011). https://doi.org/10.1090/S0025-5718-2010-02404-4

    Article  MathSciNet  MATH  Google Scholar 

  43. Ewing, R.E., Liu, M.M., Wang, J.: Superconvergence of mixed finite element approximations over quadrilaterals. SIAM J. Numer. Anal. 36(3), 772–787 (1999)

    Article  MathSciNet  Google Scholar 

  44. Falk, R.S., Neilan, M.: Stokes complexes and the construction of stable finite elements with pointwise mass conservation. SIAM J. Numer. Anal. 51(2), 1308–1326 (2013). https://doi.org/10.1137/120888132

    Article  MathSciNet  MATH  Google Scholar 

  45. Farhloul, M., Fortin, M.: A new mixed finite element for the Stokes and elasticity problems. SIAM J. Numer. Anal. 30(4), 971–990 (1993). https://doi.org/10.1137/0730051

    Article  MathSciNet  MATH  Google Scholar 

  46. Fu, G., Jin, Y., Qiu, W.: Parameter-free superconvergent \(H({\rm div})\)-conforming HDG methods for the Brinkman equations. IMA J. Numer. Anal. 39(2), 957–982 (2019). https://doi.org/10.1093/imanum/dry001

    Article  MathSciNet  MATH  Google Scholar 

  47. Gatica, G.N., Gatica, L.F., Márquez, A.: Analysis of a pseudostress-based mixed finite element method for the Brinkman model of porous media flow. Numer. Math. 126(4), 635–677 (2014). https://doi.org/10.1007/s00211-013-0577-x

    Article  MathSciNet  MATH  Google Scholar 

  48. Gatica, G.N., González, M., Meddahi, S.: A low-order mixed finite element method for a class of quasi-Newtonian Stokes flows: I: a priori error analysis. Comput. Methods Appl. Mech. Eng. 193(911), 881–892 (2004). https://doi.org/10.1016/j.cma.2003.11.007

    Article  MathSciNet  MATH  Google Scholar 

  49. Girault, V., Raviart, P.A.: Finite element methods for Navier-Stokes equations. Springer Series in Computational Mathematics. Springer-Verlag, Berlin (1986). https://doi.org/10.1007/978-3-642-61623-5. Theory and algorithms

  50. Guzmán, J., Neilan, M.: Conforming and divergence-free Stokes elements on general triangular meshes. Math. Comput. 83(285), 15–36 (2014). https://doi.org/10.1090/S0025-5718-2013-02753-6

    Article  MathSciNet  MATH  Google Scholar 

  51. Hu, J., Ma, L., Ma, R.: Optimal superconvergence analysis for the Crouzeix-Raviart and the Morley elements. Adv. Comput. Math. (2021). https://doi.org/10.1007/s10444-021-09874-7

    Article  MathSciNet  MATH  Google Scholar 

  52. Hu, J., Yu, G.: A unified analysis of quasi-optimal convergence for adaptive mixed finite element methods. SIAM J. Numer. Anal. 56(1), 296–316 (2018)

    Article  MathSciNet  Google Scholar 

  53. Huang, Y., Li, J., Wu, C.: Averaging for superconvergence: verification and application of 2d edge elements to maxwell’s equations in metamaterials. Comput. Methods Appl. Mech. Engrg. 255, 121–132 (2013)

    Article  MathSciNet  Google Scholar 

  54. Kundu, P.K., Dowling, D.R., Tryggvason, G., Cohen, I.M.: Fluid Mechanics. Academic Press, Cambridge (2015)

    Google Scholar 

  55. Li, Y.: Quasi-optimal adaptive mixed finite element methods for controlling natural norm errors. Math. Comput. 90(328), 565–593 (2021). https://doi.org/10.1090/mcom/3590

    Article  MathSciNet  MATH  Google Scholar 

  56. Li, Y.: Superconvergent flux recovery of the Rannacher-Turek nonconforming element. J. Sci. Comput. (2021). https://doi.org/10.1007/s10915-021-01445-8

    Article  MathSciNet  MATH  Google Scholar 

  57. Li, Y., Zikatanov, L.: New stabilized \({P}_1\times {P}_0\) finite element methods for nearly inviscid and incompressible flows. arXiv preprint arXiv:2109.04013 (2021)

  58. Li, Y.W.: Global superconvergence of the lowest-order mixed finite element on mildly structured meshes. SIAM J. Numer. Anal. 56(2), 792–815 (2018). https://doi.org/10.1137/17M112587X

    Article  MathSciNet  MATH  Google Scholar 

  59. Liu, H., Yan, N.: Superconvergence analysis of the nonconforming quadrilateral linear-constant scheme for Stokes equations. Adv. Comput. Math. 29(4), 375–392 (2008). https://doi.org/10.1007/s10444-007-9054-3

    Article  MathSciNet  MATH  Google Scholar 

  60. Liu, J.: Open and traction boundary conditions for the incompressible Navier-Stokes equations. J. Comput. Phys. 228(19), 7250–7267 (2009). https://doi.org/10.1016/j.jcp.2009.06.021

    Article  MathSciNet  MATH  Google Scholar 

  61. Liu, X., Li, J., Chen, Z.: A weak Galerkin finite element method for the Oseen equations. Adv. Comput. Math. 42(6), 1473–1490 (2016). https://doi.org/10.1007/s10444-016-9471-2

    Article  MathSciNet  MATH  Google Scholar 

  62. Lovadina, C., Stenberg, R.: Energy norm a posteriori error estimates for mixed finite element methods. Math. Comput. 75(256), 1659–1674 (2006). https://doi.org/10.1090/S0025-5718-06-01872-2

    Article  MathSciNet  MATH  Google Scholar 

  63. Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38(2), 466–488 (2000)

    Article  MathSciNet  Google Scholar 

  64. Mu, L., Wang, J., Ye, X.: A stable numerical algorithm for the Brinkman equations by weak Galerkin finite element methods. J. Comput. Phys. 273, 327–342 (2014). https://doi.org/10.1016/j.jcp.2014.04.017

    Article  MathSciNet  MATH  Google Scholar 

  65. Pan, J.: Global superconvergence for the bilinear-constant scheme for the Stokes problem. SIAM J. Numer. Anal. 34(6), 2424–2430 (1997). https://doi.org/10.1137/S0036142995286167

    Article  MathSciNet  MATH  Google Scholar 

  66. Park, E.J., Seo, B.: An upstream pseudostress-velocity mixed formulation for the Oseen equations. Bull. Korean Math. Soc. 51(1), 267–285 (2014). https://doi.org/10.4134/BKMS.2014.51.1.267

    Article  MathSciNet  MATH  Google Scholar 

  67. Qian, Y., Wu, S., Wang, F.: A mixed discontinuous Galerkin method with symmetric stress for Brinkman problem based on the velocity-pseudostress formulation. arXiv e-prints arXiv:1907.01246 (2019)

  68. Raviart, P.A., Thomas, J.M.: A mixed finite element method for 2nd order elliptic problems. In: Mathematical aspects of finite element methods, pp. 292–315. In Lecture Notes in Math. Vol. 606. Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome (1977)

  69. Stenberg, R.: Postprocessing schemes for some mixed finite elements. RAIRO Modél. Math. Anal. Numér. 25(1), 151–167 (1991). https://doi.org/10.1051/m2an/1991250101511

    Article  MathSciNet  MATH  Google Scholar 

  70. Temam, R.: Navier-Stokes equations: Theory and numerical analysis. In Studies in Mathematics and its Applications. North-Holland Publishing Co., Amsterdam (1977)

  71. Wang, J., Ye, X.: Superconvergence of finite element approximations for the Stokes problem by projection methods. SIAM J. Numer. Anal. 39(3), 1001–1013 (2001). https://doi.org/10.1137/S003614290037589X

    Article  MathSciNet  MATH  Google Scholar 

  72. Wang, J., Ye, X.: A weak Galerkin finite element method for the stokes equations. Adv. Comput. Math. 42(1), 155–174 (2016). https://doi.org/10.1007/s10444-015-9415-2

    Article  MathSciNet  MATH  Google Scholar 

  73. Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput. 73(247), 1139–1152 (2004). https://doi.org/10.1090/S0025-5718-03-01600-4

    Article  MathSciNet  MATH  Google Scholar 

  74. Ye, X.: Superconvergence of nonconforming finite element method for the Stokes equations. Numer. Methods Partial Differ. Equ. 18(2), 143–154 (2002). https://doi.org/10.1002/num.1036.abs

    Article  MathSciNet  MATH  Google Scholar 

  75. Zhang, Z., Naga, A.: A new finite element gradient recovery method: superconvergence property. SIAM J. Sci. Comput. 26(4), 1192–1213 (2005). https://doi.org/10.1137/S1064827503402837

    Article  MathSciNet  MATH  Google Scholar 

  76. Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Internat. J. Numer. Methods Eng. 24(2), 337–357 (1987). https://doi.org/10.1002/nme.1620240206

    Article  MathSciNet  MATH  Google Scholar 

  77. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates: I: the recovery technique. Internat. J. Numer. Methods Eng. 33(7), 1331–1364 (1992). https://doi.org/10.1002/nme.1620330702

    Article  MathSciNet  MATH  Google Scholar 

  78. Zienkiewicz, O.C., Zhu, J.Z.: The superconvergent patch recovery and a posteriori error estimates: II: error estimates and adaptivity. Internat. J. Numer. Methods Eng. 33(7), 1365–1382 (1992)

    Article  MathSciNet  Google Scholar 

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  1. Department of Engineering Science and Mechanics, The Pennsylvania State University, University Park, PA, 16802, USA

    Xi Chen

  2. Department of Mathematics, The Pennsylvania State University, University Park, PA, 16802, USA

    Yuwen Li

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Chen, X., Li, Y. Superconvergent Pseudostress-Velocity Finite Element Methods for the Oseen Equations. J Sci Comput 92, 17 (2022). https://doi.org/10.1007/s10915-022-01856-1

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