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An a posteriori error estimate for a dual mixed method applied to Stokes system with non-null source terms

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Abstract

In this work, we focus our attention in the Stokes flow with nonhomogeneous source terms, formulated in dual mixed form. For the sake of completeness, we begin recalling the corresponding well-posedness at continuous and discrete levels. After that, and with the help of a kind of a quasi-Helmholtz decomposition of functions in H(div), we develop a residual type a posteriori error analysis, deducing an estimator that is reliable and locally efficient. Finally, we provide numerical experiments, which confirm our theoretical results on the a posteriori error estimator and illustrate the performance of the corresponding adaptive algorithm, supporting its use in practice.

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References

  1. Agmon, S.: Lectures on elliptic boundary value problems. Van Nostrand Mathematical Studies, No. 2. D. Van Nostrand Co. Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. (1965)

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

    Article  MathSciNet  Google Scholar 

  3. Barrios, T.P., Behrens, E.M., Bustinza, R.: A stabilized mixed method applied to Stokes system with nonhomogeneous source terms: the stationary case. Dedicated to Prof. R. Rodríguez, on the occasion of his 65th birthday. Internat. J. Numer. Methods Fluids 92(6), 509–527 (2020)

    Article  MathSciNet  Google Scholar 

  4. Barrios, T.P., Behrens, E.M., González, M.: Low cost a posteriori error estimators for an augmented mixed FEM in linear elasticity. Appl. Numer Math. 84, 46–65 (2014)

    Article  MathSciNet  Google Scholar 

  5. Barrios, T.P., Bustinza, R., Campos, C.: A note on a posteriori error estimate for dual mixed method. Departamento de Ingeniería Matemática. Universidad de Concepción, Chile, Pre-print 2020-08 (2020)

  6. Barrios, T.P., Bustinza, R., García, G.C., González, M.: An a posteriori error analysis of a velocity-pseudostress formulation of the generalized Stokes problem. J. Comput. Appl. Math. 357, 349–365 (2019)

    Article  MathSciNet  Google Scholar 

  7. Barrios, T.P., Bustinza, R., García, G.C., Hernández, E.: On stabilized mixed methods for generalized Stokes problem based on the velocity-pseudostress formulation: a priori error estimates. Comput. Methods Appl. Mech. Engrg. 237–240, 78–87 (2012)

    Article  MathSciNet  Google Scholar 

  8. Brezzi, F., Fortin, M.: Mixed and Hybrid Finite Element Methods, Volume 15 of Springer Series in Computational Mathematics. Springer, New York (1991)

    Book  Google Scholar 

  9. Cai, Z., Lee, B., Wang, P.: Least-squares methods for incompressible Newtonian fluid flow: linear stationary problems. SIAM J. Numer. Anal. 42(2), 843–859 (2004)

    Article  MathSciNet  Google Scholar 

  10. 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)

    Article  MathSciNet  Google Scholar 

  11. Carstensen, C., Bartels, S., Jansche, S.: A posteriori error estimates for nonconforming finite element methods. Numer. Math. 92(2), 233–256 (2002)

    Article  MathSciNet  Google Scholar 

  12. 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 

  13. Ciarlet, P.G.: The finite element method for elliptic problems, volume 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM), Philadelphia. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)] (2002)

  14. Clément, P.: Approximation by finite element functions using local regularization. Rev. Française Automat. Informat. Recherche Opér. Sér. 9(R-2), 77–84 (1975)

    MathSciNet  MATH  Google Scholar 

  15. Durán, R., Gastaldi, L., Lombardi, A.: Analysis of finite element approximations of S,tokes equations with nonsmooth data. SIAM J. Numer Anal. 58(6), 3309–3331 (2020)

    Article  MathSciNet  Google Scholar 

  16. Farhloul, M., Fortin, M.: A new mixed finite element for the Stokes and elasticity problems. SIAM J. Numer. Anal. 30(4), 971–990 (1993)

    Article  MathSciNet  Google Scholar 

  17. Farhloul, M., Nicaise, S., Paquet, L.: A posteriori error estimation for the dual mixed finite element method of the Stokes problem. Numer. Funct. Anal Optim. 27(7-8), 831–846 (2006)

    Article  MathSciNet  Google Scholar 

  18. Figueroa, L.E., Gatica, G.N., Márquez, A.: Augmented mixed finite element methods for the stationary Stokes equations. SIAM J. Sci. Comput. 31(2):1082–1119 (2008/09)

  19. Gatica, G.N., Márquez, A., Sánchez, M.A.: Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equations. Comput. Methods Appl. Mech Eng. 199(17-20), 1064–1079 (2010)

    Article  MathSciNet  Google Scholar 

  20. Gatica, G.N., Márquez, A., Sánchez, M.A.: A priori and a posteriori error analyses of a velocity-pseudostress formulation for a class of quasi-Newtonian Stokes flows. Comput. Methods Appl. Mech. Engrg. 200(17-20), 1619–1636 (2011)

    Article  MathSciNet  Google Scholar 

  21. Girault, V., Raviart, P.-A.: Finite Element Methods for Navier-Stokes Equations, Volume 5 of Springer Series in Computational Mathematics. Springer, Berlin. Theory and algorithms (1986)

  22. Hiptmair, R.: Finite elements in computational electromagnetism. Acta Numer. 11, 237–339 (2002)

    Article  MathSciNet  Google Scholar 

  23. Kovasznay, L.I.G.: Laminar flow behind two-dimensional grid. Proc. Cambridge Philos. Soc. 44, 58–62 (1948)

    Article  MathSciNet  Google Scholar 

  24. Roberts, J.E., Thomas, J.-M.: Mixed and hybrid methods. In Handbook of numerical analysis, Vol. II, Handb. Numer. Anal., II, pp. 523–639. North-Holland (1991)

  25. Verfürth, R.: A posteriori error estimation and adaptive mesh-refinement techniques. In: Proceedings of the F,ifth International Congress on Computational and Applied Mathematics (Leuven, 1992), vol. 50, pp. 67–83 (1994)

  26. Verfürth, R.: A Review of a Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Advances in Numerical Mathematics. Wiley-Teubner, Chichester (1996)

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Funding

This research has been partially supported by ANID-Chile through the project Centro de Modelamiento Matemático (AFB170001) of the PIA Program: Concurso Apoyo a Centros Científicos y Tecnológicos de Excelencia con Financiamiento Basal CONICYT-Chile and by FONDECYT Grant 1200051; by Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción (Chile), and by Dirección de Investigación de la Universidad Católica de la Santísima Concepción (Chile), through Incentivo Mensual program.

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Authors and Affiliations

  1. Departamento de Matemática y Física Aplicadas, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile

    Tomás P. Barrios

  2. Departamento de Ingeniería Civil, Universidad Católica de la Santísima Concepción, Casilla 297, Concepción, Chile

    Edwin M. Behrens

  3. Centro de Investigación en Ingeniería Matemática (CI2MA) & Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Rommel Bustinza

Authors
  1. Tomás P. Barrios
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  2. Edwin M. Behrens
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  3. Rommel Bustinza
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All authors have contributed equally in this article.

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Correspondence to Rommel Bustinza.

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Communicated by: Paul Houston

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Barrios, T.P., Behrens, E.M. & Bustinza, R. An a posteriori error estimate for a dual mixed method applied to Stokes system with non-null source terms. Adv Comput Math 47, 77 (2021). https://doi.org/10.1007/s10444-021-09906-2

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