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Error Analysis of a B-Spline Based Finite-Element Method for Modeling Wind-Driven Ocean Circulation

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Abstract

We present the results of an error analysis of a B-spline based finite-element approximation of the stream-function formulation of the large scale wind-driven ocean circulation. In particular, we derive optimal error estimates for h-refinement using a Nitsche-type variational formulations of the two simplied linear models of the stationary quasigeostrophic equations, namely the Stommel and Stommel–Munk models. Numerical results obtained from simulations performed on rectangular and embedded geometries confirm the error analysis.

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References

  1. Vallis, G .K.: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-scale Circulation. Cambridge University Press, Cambridge (2006)

    Book  Google Scholar 

  2. Cushman-Roisin, B., Beckers, J .M.: Introduction to Geophysical Fluid Dynamics: Physical and Numerical Aspects. Academic Press, Cambridge (2011)

    MATH  Google Scholar 

  3. Majda, A.: Introduction to PDEs and Waves for the Atmosphere and Ocean, volume 9 of Courant Lecture Notes in Mathematics. American Mathematical Society, Providence (2003)

  4. Majda, A., Wang, X.: Nonlinear Dynamics and Statistical Theories for Basic Geophysical Flows. Cambridge University Press, Cambridge (2006)

    Book  MATH  Google Scholar 

  5. McWilliams, J .C.: Fundamentals of Geophysical Fluid Dynamics. Cambridge University Press, Cambridge (2006)

    MATH  Google Scholar 

  6. Pedlosky, J.: Geophysical Fluid Dynamics. Springer, Berlin (1992)

    MATH  Google Scholar 

  7. Fix, G.J.: Finite element models for ocean circulation problems. SIAM J. Appl. Math. 29(3), 371–387 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  8. Foster, E.L., Iliescu, T., Wang, Z.: A finite element discretization of the streamfunction formulation of the stationary quasi-geostrophic equations of the ocean. Comput. Methods Appl. Mech. Eng. 261–262, 105–117 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  9. Kim, T.-Y., Iliescu, T., Fried, E.: B-spline based finite-element method for the stationary quasi-geostrophic equations of the ocean. Comput. Methods Appl. Mech. Eng. 286, 168–191 (2015)

    Article  MathSciNet  Google Scholar 

  10. Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194(39–41), 4135–4195 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Heltai, L., Arroyo, M., DeSimone, A.: Nonsingular isogeometric boundary element method for stokes flows in 3D. Comput. Methods Appl. Mech. Eng. 268, 514–539 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Manzoni, A., Salmoiraghi, F., Heltai, L.: Reduced Basis Isogeometric Methods (RB-IGA) for the real-time simulation of potential flows about parametrized NACA airfoils. Comput. Methods Appl. Mech. Eng. 284, 1147–1180 (2015)

    Article  MathSciNet  Google Scholar 

  13. Bazilevs, Y., Hughes, T.J.R.: Weak imposition of Dirichlet boundary conditions in fluid mechanics. Comput. Fluids 36(1), 12–26 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Nitsche, J.: Uber ein variationsprinzip zur lösung von dirichlet-problemen bei verwendung von teilräumen, die keinen randbedingungen unterworfen sind. In: Abhandlungen aus dem mathematischen Seminar der Universität Hamburg, vol.  36, pp. 9–15. Springer, Berlin (1971)

  15. Embar, A., Dolbow, J., Harari, I.: Imposing Dirichlet boundary conditions with Nitsche’s method and spline-based finite elements. Int. J. Numer. Methods Eng. 83(7), 877–898 (2010)

    MathSciNet  MATH  Google Scholar 

  16. Kim, T.-Y., Dolbow, J.E.: An edge-bubble stabilized finite element method for fourth-order parabolic problems. Finite Elem. Anal. Des. 45(8), 485–494 (2009)

    Article  MathSciNet  Google Scholar 

  17. Kim, T.-Y., Dolbow, J., Fried, E.: A numerical method for a second-gradient theory of incompressible fluid flow. J. Comput. Phys. 223(2), 551–570 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kim, Tae-Yeon, Dolbow, John E., Fried, Eliot: Numerical study of the grain-size dependent Young’s modulus and Poisson’s ratio of bulk nanocrystalline materials. Int. J. Solids Struct. 49(26), 3942–3952 (2012)

    Article  Google Scholar 

  19. Kim, T.-Y., Puntel, E., Fried, E.: Numerical study of the wrinkling of a stretched thin sheet. Int. J. Solids Struct. 49(5), 771–782 (2012)

    Article  Google Scholar 

  20. Fernández-Méndez, S., Huerta, A.: Imposing essential boundary conditions in mesh-free methods. Comput. Methods Appl. Mech. Eng. 193(12), 1257–1275 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Dolbow, J., Harari, I.: An efficient finite element method for embedded interface problems. Int. J. Numer. Methods Eng. 78(2), 229–252 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  22. Hansbo, A., Hansbo, P.: An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems. Comput. Methods Appl. Mech. Eng. 191(47–48), 5537–5552 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Kamensky, D., Hsu, M.-C., Schillinger, Dk, Evans, J.A., Aggarwal, A., Bazilevs, Y., Sacks, M.S., Hughes, T.J.R.: An immersogeometric variational framework for fluid-structure interaction. Comput. Methods Appl. Mech. Eng. 284, 1005–1053 (2015)

    Article  MathSciNet  Google Scholar 

  24. Jiang, W., Annavarapu, C., Dolbow, J.E., Harari, I.: A robust Nitsche’s formulation for interface problems with spline-based finite elements. Int. J. Numer. Methods Eng. 104(7), 676–696 (2015)

    Article  MathSciNet  Google Scholar 

  25. Boffi, D., Gastaldi, L., Heltai, L., Peskin, C.S.: On the hyper-elastic formulation of the immersed boundary method. Comput. Methods Appl. Mech. Eng. 197(25–28), 2210–2231 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Cottrell, J .A., Hughes, T .J .R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Hoboken (2009)

    Book  Google Scholar 

  27. Piegl, L., Tiller, W.: The NURBS Book, 2nd edn. Springer, Berlin (1997)

    Book  MATH  Google Scholar 

  28. Beirão da Veiga, L., Buffa, A., Rivas, J., Sangalli, G.: Some estimates for h-p-k-refinement in isogeometric analysis. Numerische Mathematik 118(2), 271–305 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  29. Bazilevs, Y., Beirão da Veiga, L., Cottrell, J.A., Hughes, T.J.R., Sangalli, G.: Isogeometric analysis: approximation, stability and error estimates for h-refined meshes. Math. Models Methods Appl. Sci. 16(07), 1031–1090 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  30. Brenner, S .C., Scott, R.: The Mathematical Theory of Finite Element Methods. Springer, Berlin (2007)

    MATH  Google Scholar 

  31. Ciarlet, P .G.: The Finite Element Method for Elliptic Problems. SIAM, Pathum Wan (2002)

    Book  MATH  Google Scholar 

  32. Engel, G., Garikipati, K., Hughes, T.J.R., Larson, M.G., Mazzei, L., Taylor, R.L.: Continuous/discontinuous finite element approximations of fourth-order elliptic problems in structural and continuum mechanics with applications to thin beams and plates, and strain gradient elasticity. Comput. Methods Appl. Mech. Eng. 191(34), 3669–3750 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19(4), 742–760 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  34. Bazilevs, Y., Michler, C., Calo, V.M., Hughes, T.J.R.: Isogeometric variational multiscale modeling of wall-bounded turbulent flows with weakly enforced boundary conditions on unstretched meshes. Comput. Methods Appl. Mech. Eng. 199(13), 780–790 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  35. Bramble, J.H., Dupont, T., Thomée, Vidar: Projection methods for Dirichlet’s problem in approximating polygonal domains with boundary-value corrections. Math. Comput. 26(120), 869–879 (1972)

    MATH  Google Scholar 

  36. Dréau, K., Chevaugeon, N., Moës, N.: Studied X-FEM enrichment to handle material interfaces with higher order finite element. Comput. Methods Appl. Mech. Eng. 199(29–32), 1922–1936 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  37. Ruess, M., Schillinger, D., Bazilevs, Y., Varduhn, V., Rank, E.: Weakly enforced essential boundary conditions for NURBS-embedded and trimmed NURBS geometries on the basis of the finite cell method. Int. J. Numer. Methods Eng. 95(10), 811–846 (2013)

    Article  MathSciNet  Google Scholar 

  38. Myers, P.G., Weaver, A.J.: A diagnostic barotropic finite-element ocean circulation model. J. Atmos. Ocean. Technol. 12(3), 511–526 (1995)

    Article  Google Scholar 

  39. Cascon, J.M., Garcia, G.C., Rodriguez, R.: A priori and a posteriori error analysis for a large-scale ocean circulation finite element model. Comput. Methods Appl. Mech. Eng. 192(51), 5305–5327 (2003)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The work has been partially supported by (N.R.) ERC-2010-AdG No. 267802 Analysis of Multiscale Systems Driven by Functionals, and by (L.H.) project OpenViewSHIP, “Sviluppo di un ecosistema computazionale per la progettazione idrodinamica del sistema elica-carena”, supported by Regione FVG - PAR FSC 2007–2013, Fondo per lo Sviluppo e la Coesione and by the project “TRIM – Tecnologia e Ricerca Industriale per la Mobilità Marina”, CTN01-00176-163601, supported by MIUR, the Italian Ministry of Instruction, University and Research. E.F. gratefully acknowledges support from the Okinawa Institute of Science and Technology Graduate University with subsidy funding from the Cabinet Office, Government of Japan.

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Rotundo, N., Kim, TY., Jiang, W. et al. Error Analysis of a B-Spline Based Finite-Element Method for Modeling Wind-Driven Ocean Circulation. J Sci Comput 69, 430–459 (2016). https://doi.org/10.1007/s10915-016-0201-1

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  • DOI: https://doi.org/10.1007/s10915-016-0201-1

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