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A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem

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Abstract

In this paper, we develop an a posteriori error analysis of a mixed finite element method for a fluid–solid interaction problem posed in the plane. The media are governed by the acoustic and elastodynamic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the normal displacements of the solid and the fluid. The coupling of primal and dual-mixed finite element methods is applied to compute both the pressure of the scattered wave in the linearized fluid and the elastic vibrations that take place in the elastic body. The finite element subspaces consider continuous piecewise linear elements for the pressure and a Lagrange multiplier defined on the interface, and PEERS for the stress and rotation in the solid domain. We derive a reliable and efficient residual-based a posteriori error estimator for this coupled problem. Suitable auxiliary problems, the continuous inf-sup conditions satisfied by the bilinear forms involved, a discrete Helmholtz decomposition, and the local approximation properties of the Clément interpolant and Raviart–Thomas operator are the main tools for proving the reliability of the estimator. Then, Helmholtz decomposition, inverse inequalities, and the localization technique based on triangle-bubble and edge-bubble functions are employed to show the efficiency. Finally, some numerical results confirming the reliability and efficiency of the estimator are reported.

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References

  1. Agmon S.: Lectures on Elliptic Boundary Value Problems. Van Nostrand, Princeton (1965)

    MATH  Google Scholar 

  2. Alonso A.: Error estimators for a mixed method. Numer. Math. 74, 385–395 (1996)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  4. Arnold D.N., Brezzi F., Douglas J.: PEERS: A new mixed finite element method for plane elasticity. Jpn. J. Appl. Math. 1, 347–367 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  5. Babuška I., Ihlenburg F., Strouboulis T., Gangaraj S.K.: A posteriori error estimation for finite element solutions of Helmholtz’ equation. Part I. The quality of local indicators and estimators. Int. J. Numer. Methods Eng. 40(18), 3443–3462 (1997)

    Article  MATH  Google Scholar 

  6. Babuška I., Ihlenburg F., Strouboulis T., Gangaraj S.K.: A posteriori error estimation for finite element solutions of Helmholtz’ equation – Part II. Estimation of the pollution error. Int. J. Numer. Methods Eng. 40(21), 3883–3900 (1997)

    Article  MATH  Google Scholar 

  7. Barrios T., Gatica G.N., González M., Heuer N.: A residual based a posteriori error estimator for an augmented mixed finite element method in linear elasticity. ESAIM Math. Model. Numer. Anal. 40(5), 843–869 (2006)

    Article  MATH  Google Scholar 

  8. Bernardi C., Canuto C., Maday Y.: Generalized inf-sup conditions for Chebyshev spectral approximations of the Stokes problem. SIAM J. Numer. Anal. 25(6), 1237–1271 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bielak J., MacCamy R.C.: Symmetric finite element and boundary integral coupling methods for fluid–solid interaction. Q. Appl. Math. 49, 107–119 (1991)

    MATH  MathSciNet  Google Scholar 

  10. Bouillard Ph.: Influence of the pollution on the admissible field error estimation for FE solutions of the Helmholtz equation. Int. J. Numer. Methods Eng. 45(7), 783–800 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  11. Braess D., Verfürth R.: A posteriori error estimators for the Raviart–Thomas element. SIAM J. Numer. Anal. 33, 2431–2444 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  12. Brezzi F., Fortin M.: Mixed and Hybrid Finite Element Methods. Springer, New York (1991)

    MATH  Google Scholar 

  13. Buffa A.: Remarks on the discretization of some non-coercive operator with applications to heterogeneous Maxwell equations. SIAM J. Numer. Anal. 43(1), 1–18 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  14. Carstensen C.: An a posteriori error estimate for a first kind integral equation. Math. Comput. 66(217), 139–155 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  15. Carstensen C.: A posteriori error estimate for the mixed finite element method. Math. Comput. 66(218), 465–476 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  16. Carstensen C., Dolzmann G.: A posteriori error estimates for mixed FEM in elasticity. Numer. Math. 81, 187–209 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  17. Ciarlet P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)

    MATH  Google Scholar 

  18. Clément P.: Approximation by finite element functions using local regularisation. RAIRO Model. Math. Anal. Numer. 9, 77–84 (1975)

    Google Scholar 

  19. Colton D., Kress R.: Inverse Acoustic and Electromagnetic Scattering Theory, 2nd edn. Springer, Berlin (1998)

    MATH  Google Scholar 

  20. Gatica G.N.: A note on the efficiency of residual-based a-posteriori error estimators for some mixed finite element methods. Electron. Trans. Numer. Anal. 17, 218–233 (2004)

    MATH  MathSciNet  Google Scholar 

  21. Gatica G.N., Maischak M.: A posteriori error estimates for the mixed finite element method with Lagrange multipliers. Numer. Methods Partial Differ. Equ. 21(3), 421–450 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  22. Gatica G.N., Márquez A., Meddahi S.: Analysis of the coupling of primal and dual-mixed finite element methods for a two-dimensional fluid–solid interaction problem. SIAM J. Numer. Anal. 45(5), 2072–2097 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  23. Gatica, G.N., Márquez, A., Meddahi, S.: Analysis of the coupling of BEM, FEM, and mixed-FEM for a two-dimensional fluid–solid interaction problem. Appl. Numer. Math. (2009). doi:10.1016/j.apnum.2008.12.025

  24. Gatica G.N., Meddahi S.: An a-posteriori error estimate for the coupling of BEM and mixed-FEM. Numer. Funct. Anal. Optim. 20(5–6), 449–472 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  25. Hsiao, G.C.: The coupling of BEM and FEM—a brief review. In: Boundary Elements X, vol. 1, pp. 431–445. Springer, New York (1988)

  26. Hsiao G.C.: On the boundary-field equation methods for fluid-structure interactions. In: Jentsch, L., Tröltzsch, F. (eds) Problems and Methods in Mathematical Physics, Teubner-Text zur Mathematik, Band 34, pp. 79–88. B. G. Teubner Veriagsgesellschaft, Stuttgart (1994)

    Google Scholar 

  27. Hsiao G.C., Kleinman R.E., Li R.-X., van der Berg P.M.: Residual error—a simple and sufficient estimate of actual error in solutions of boundary integral equations. In: Grill, S., Brebbia, C.A., Cheng, A.H.-D. (eds) Proceedings of the Fifth International Conference on Boundary Element Technology, vol. 1, pp. 73–83. Computational Mechanics Publications, Boston (1990)

    Google Scholar 

  28. Hsiao G.C., Kleinman R.E., Roach G.F.: Weak solutions of fluid–solid interaction problems. Math. Nachr. 218, 139–163 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. Hsiao G.C., Kleinman R.E., Schuetz L.S.: On variational formulations of boundary value problems for fluid–solid interactions. In: McCarthy, M.F., Hayes, M.A. (eds) Elastic Wave Propagation, pp. 321–326. Elsevier, North-Holland (1989)

    Google Scholar 

  30. Ihlenburg F.: Finite Element Analysis of Acoustic Scattering. Springer, New York (1998)

    Book  MATH  Google Scholar 

  31. Ihlenburg F., Babuška I.: Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation. Int. J. Numer. Methods Eng. 38(22), 3745–3774 (1995)

    Article  MATH  Google Scholar 

  32. Irimie S., Boiullard Ph.: A reliable a posteriori error estimator for the finite element solution of the Helmholtz equation. Comput. Methods Appl. Mech. Eng. 190(31), 4027–4042 (2001)

    Article  MATH  Google Scholar 

  33. Kress R.: Linear Integral Equations. Springer, Berlin (1989)

    MATH  Google Scholar 

  34. Lonsing M., Verfürth R.: A posteriori error estimators for mixed finite element methods in linear elasticity. Numer. Math. 97(4), 757–778 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  35. Lovadina C., Stenberg R.: Energy norm a posteriori error estimates for mixed finite element methods. Math. Comput. 75(256), 1659–1674 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  36. Meddahi S., Márquez A., Selgas V.: Computing acoustic waves in an inhomogeneous medium of the plane by a coupling of spectral and finite elements. SIAM J. Numer. Anal. 41, 1729–1750 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  37. Meddahi S., Márquez A., Selgas V.: A new BEM-FEM coupling strategy for two-dimensional fluid–solid interaction problems. J. Comput. Phys. 199, 205–220 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  38. Meddahi S., Sayas F.-J.: Analysis of a new BEM-FEM coupling for two dimensional fluid–solid interaction. Numer. Methods Partial Differ. Equ. 21(6), 1017–1042 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  39. Repin S., Sauter S., Smolianski A.: Two-sided a posteriori error estimates for mixed formulations of elliptic problems. SIAM J. Numer. Anal. 45(3), 928–945 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  40. Roberts, J.E., Thomas, J.M.: Mixed and Hybrid Methods. In: Ciarlet, P.G., Lions, J.L. (eds.) Handbook of Numerical Analysis. Finite Element Methods (Part 1), vol. II. North-Holland, Amsterdam (1991)

  41. Stephan E.P.: Coupling of boundary element methods and finite element methods. In: Stein, E., Borst, R., Hughes, T.J.R. (eds) Encyclopedia of Computational Mechanics, vol. 1, pp. 375–412. Wiley, New York (2004)

    Google Scholar 

  42. Stewart J.R., Hughes T.J.R.: A posteriori error estimation and adaptive finite element computation of the Helmholtz equation in exterior domains. Finite Elem. Anal. Des. 22(1), 15–24 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  43. Stewart J.R., Hughes T.J.R.: An a posteriori error estimator and hp-adaptive strategy for finite element discretizations of the Helmholtz equation in exterior domains. Finite Elem. Anal. Des. 25(1–2), 1–26 (1996)

    MathSciNet  Google Scholar 

  44. Stewart J.R., Hughes T.J.R.: Explicit residual-based a posteriori error estimation for finite element discretizations of the Helmholtz equation: Computation of the constant and new measures of error estimator quality. Comput. Methods Appl. Mech. Eng. 131(3–4), 335–363 (1997)

    MathSciNet  Google Scholar 

  45. Verfürth R.: A posteriori error estimation and adaptive mesh-refinement techniques. J. Comput. Appl. Math. 50, 67–83 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  46. Verfürth R.: A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley, Chichester (1996)

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. CI2MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Gabriel N. Gatica

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    George C. Hsiao

  3. Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, Oviedo, Spain

    Salim Meddahi

Authors
  1. Gabriel N. Gatica
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  2. George C. Hsiao
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  3. Salim Meddahi
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Corresponding author

Correspondence to Gabriel N. Gatica.

Additional information

This research was partially supported by FONDAP and BASAL projects CMM, Universidad de Chile, by Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción, and by Ministery of Education and Science of Spain through the Project MTM2007-65088.

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Gatica, G.N., Hsiao, G.C. & Meddahi, S. A residual-based a posteriori error estimator for a two-dimensional fluid–solid interaction problem. Numer. Math. 114, 63–106 (2009). https://doi.org/10.1007/s00211-009-0250-6

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  • DOI: https://doi.org/10.1007/s00211-009-0250-6

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