A priori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem

Carolina Domínguez; Gabriel N. Gatica; Salim Meddahi; Ricardo Oyarzúa

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2013)

  • Volume: 47, Issue: 2, page 471-506
  • ISSN: 0764-583X

Abstract

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We introduce and analyze a fully-mixed finite element method for a fluid-solid interaction problem in 2D. The model consists of an elastic body which is subject to a given incident wave that travels in the fluid surrounding it. Actually, the fluid is supposed to occupy an annular region, and hence a Robin boundary condition imitating the behavior of the scattered field at infinity is imposed on its exterior boundary, which is located far from the obstacle. The media are governed by the elastodynamic and acoustic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements. We first apply dual-mixed approaches in both domains, and then employ the governing equations to eliminate the displacement u of the solid and the pressure p of the fluid. In addition, since both transmission conditions become essential, they are enforced weakly by means of two suitable Lagrange multipliers. As a consequence, the Cauchy stress tensor and the rotation of the solid, together with the gradient of p and the traces of u and p on the boundary of the fluid, constitute the unknowns of the coupled problem. Next, we show that suitable decompositions of the spaces to which the stress and the gradient of p belong, allow the application of the Babuška–Brezzi theory and the Fredholm alternative for analyzing the solvability of the resulting continuous formulation. The unknowns of the solid and the fluid are then approximated by a conforming Galerkin scheme defined in terms of PEERS elements in the solid, Raviart–Thomas of lowest order in the fluid, and continuous piecewise linear functions on the boundary. Then, the analysis of the discrete method relies on a stable decomposition of the corresponding finite element spaces and also on a classical result on projection methods for Fredholm operators of index zero. Finally, some numerical results illustrating the theory are presented.

How to cite

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Domínguez, Carolina, et al. "A priori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 47.2 (2013): 471-506. <http://eudml.org/doc/273300>.

@article{Domínguez2013,
abstract = {We introduce and analyze a fully-mixed finite element method for a fluid-solid interaction problem in 2D. The model consists of an elastic body which is subject to a given incident wave that travels in the fluid surrounding it. Actually, the fluid is supposed to occupy an annular region, and hence a Robin boundary condition imitating the behavior of the scattered field at infinity is imposed on its exterior boundary, which is located far from the obstacle. The media are governed by the elastodynamic and acoustic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements. We first apply dual-mixed approaches in both domains, and then employ the governing equations to eliminate the displacement u of the solid and the pressure p of the fluid. In addition, since both transmission conditions become essential, they are enforced weakly by means of two suitable Lagrange multipliers. As a consequence, the Cauchy stress tensor and the rotation of the solid, together with the gradient of p and the traces of u and p on the boundary of the fluid, constitute the unknowns of the coupled problem. Next, we show that suitable decompositions of the spaces to which the stress and the gradient of p belong, allow the application of the Babuška–Brezzi theory and the Fredholm alternative for analyzing the solvability of the resulting continuous formulation. The unknowns of the solid and the fluid are then approximated by a conforming Galerkin scheme defined in terms of PEERS elements in the solid, Raviart–Thomas of lowest order in the fluid, and continuous piecewise linear functions on the boundary. Then, the analysis of the discrete method relies on a stable decomposition of the corresponding finite element spaces and also on a classical result on projection methods for Fredholm operators of index zero. Finally, some numerical results illustrating the theory are presented.},
author = {Domínguez, Carolina, Gatica, Gabriel N., Meddahi, Salim, Oyarzúa, Ricardo},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {mixed finite elements; Helmholtz equation; elastodynamic equation},
language = {eng},
number = {2},
pages = {471-506},
publisher = {EDP-Sciences},
title = {A priori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem},
url = {http://eudml.org/doc/273300},
volume = {47},
year = {2013},
}

TY - JOUR
AU - Domínguez, Carolina
AU - Gatica, Gabriel N.
AU - Meddahi, Salim
AU - Oyarzúa, Ricardo
TI - A priori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2013
PB - EDP-Sciences
VL - 47
IS - 2
SP - 471
EP - 506
AB - We introduce and analyze a fully-mixed finite element method for a fluid-solid interaction problem in 2D. The model consists of an elastic body which is subject to a given incident wave that travels in the fluid surrounding it. Actually, the fluid is supposed to occupy an annular region, and hence a Robin boundary condition imitating the behavior of the scattered field at infinity is imposed on its exterior boundary, which is located far from the obstacle. The media are governed by the elastodynamic and acoustic equations in time-harmonic regime, respectively, and the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements. We first apply dual-mixed approaches in both domains, and then employ the governing equations to eliminate the displacement u of the solid and the pressure p of the fluid. In addition, since both transmission conditions become essential, they are enforced weakly by means of two suitable Lagrange multipliers. As a consequence, the Cauchy stress tensor and the rotation of the solid, together with the gradient of p and the traces of u and p on the boundary of the fluid, constitute the unknowns of the coupled problem. Next, we show that suitable decompositions of the spaces to which the stress and the gradient of p belong, allow the application of the Babuška–Brezzi theory and the Fredholm alternative for analyzing the solvability of the resulting continuous formulation. The unknowns of the solid and the fluid are then approximated by a conforming Galerkin scheme defined in terms of PEERS elements in the solid, Raviart–Thomas of lowest order in the fluid, and continuous piecewise linear functions on the boundary. Then, the analysis of the discrete method relies on a stable decomposition of the corresponding finite element spaces and also on a classical result on projection methods for Fredholm operators of index zero. Finally, some numerical results illustrating the theory are presented.
LA - eng
KW - mixed finite elements; Helmholtz equation; elastodynamic equation
UR - http://eudml.org/doc/273300
ER -

References

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  1. [1] D.N. Arnold, F. Brezzi and J. Douglas, PEERS : A new mixed finite element method for plane elasticity. Japan J. Appl. Math.1 (1984) 347–367. Zbl0633.73074MR840802
  2. [2] I. Babuška and A.K. Aziz, Survey lectures on the mathematical foundations of the finite element method. in The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, edited by A.K. Aziz. Academic Press, New York (1972). Zbl0268.65052MR421106
  3. [3] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag New York, Inc. (1994). Zbl1135.65042MR1278258
  4. [4] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag (1991). Zbl0788.73002MR1115205
  5. [5] J. Bielak and R.C. MacCamy, Symmetric finite element and boundary integral coupling methods for fluid-solid interaction. Quarterly Appl. Math.49 (1991) 107–119. Zbl0731.76043MR1096235
  6. [6] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edition. Springer-Verlag, Berlin (1998). Zbl0760.35053MR1635980
  7. [7] G.N. Gatica, A. Márquez and S. Meddahi, 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 (2007) 2072–2097. Zbl1225.74087MR2346371
  8. [8] G.N. Gatica, A. Márquez and S. Meddahi, A new dual-mixed finite element method for the plane linear elasticity problem with pure traction boundary conditions. Comput. Methods Appl. Mech. Engrg.197 (2008) 1115–1130. Zbl1169.74601MR2376978
  9. [9] G.N. Gatica, A. Márquez and S. Meddahi, Analysis of the coupling of BEM, FEM and mixed-FEM for a two-dimensional fluid-solid interaction problem. Appl. Numer. Math. 59 (2009) 2735–2750. Zbl1171.76027MR2566767
  10. [10] G.N. Gatica, A. Márquez and S. Meddahi, Analysis of the coupling of Lagrange and Arnold-Falk-Winther finite elements for a fluid-solid interaction problem in 3D. SIAM J. Numer. Anal.50 (2012) 1648–1674. Zbl06070634MR2970759
  11. [11] G.N. Gatica, A. Márquez and S. Meddahi, Analysis of an augmented fully-mixed finite element method for a three-dimensional fluid-solid interaction problem. Preprint 2011-23, Centro de Investigación en Ingeniería Matemática (CI2MA), Universidad de Concepción (2011). MR3218341
  12. [12] G.N. Gatica, R. Oyarzúa and F.J. Sayas, Analysis of fully-mixed finite element methods for the Stokes-Darcy coupled problem. Math. Comput. 80 276 (2011) 1911–1948. Zbl1301.76047MR2813344
  13. [13] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms. Springer-Verlag. Springer Ser. Comput. Math. 5 (1986). Zbl0585.65077MR851383
  14. [14] P. Grisvard, Elliptic Problems in Non-Smooth Domains. Pitman. Monogr. Studies Math. 24 (1985). Zbl0695.35060
  15. [15] P. Grisvard, Problèmes aux limites dans les polygones. Mode d’emploi. EDF. Bulletin de la Direction des Etudes et Recherches (Serie C) 1 (1986) 21–59. Zbl0623.35031MR840970
  16. [16] R. Hiptmair, Finite elements in computational electromagnetism. Acta Numer.11 (2002) 237–339. Zbl1123.78320MR2009375
  17. [17] G.C. Hsiao, On the boundary-field equation methods for fluid-structure interactions, edited by L. Jentsch and F. Tröltzsch, Teubner-Text zur Mathematik, Band, B.G. Teubner Veriagsgesellschaft, Stuttgart, in Probl. Methods Math. Phys. 34 (1994) 79–88. Zbl0849.76040MR1288317
  18. [18] G.C. Hsiao, R.E. Kleinman and G.F. Roach, Weak solutions of fluid-solid interaction problems. Math. Nachrichten218 (2000) 139–163. Zbl0963.35043MR1784639
  19. [19] G.C. Hsiao, R.E. Kleinman and L.S. Schuetz, On variational formulations of boundary value problems for fluid-solid interactions, edited by M.F. McCarthy and M.A. Hayes. Elsevier Science Publishers B.V. (North-Holland), in Elastic Wave Propagation (1989) 321–326. MR1000990
  20. [20] F. Ihlenburg, Finite Element Analysis of Acoustic Scattering. Springer-Verlag, New York (1998). Zbl0908.65091MR1639879
  21. [21] R. Kress, Linear Integral Equ. Springer-Verlag, Berlin (1989). MR1007594
  22. [22] M. Lonsing and R. Verfürth, On the stability of BDMS and PEERS elements. Numer. Math.99 (2004) 131–140. Zbl1076.65090MR2101787
  23. [23] A. Márquez, S. Meddahi and V. Selgas, A new BEM-FEM coupling strategy for two-dimensional fluid-solid interaction problems. J. Comput. Phys.199 (2004) 205–220. Zbl1127.74328MR2081003
  24. [24] W. McLean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000). Zbl0948.35001MR1742312
  25. [25] S. Meddahi and F.-J. Sayas, Analysis of a new BEM-FEM coupling for two dimensional fluid-solid interaction. Numer. Methods Partial Differ. Equ.21 (2005) 1017–1042. Zbl1078.74054MR2169165
  26. [26] J.E. Roberts and J.M. Thomas, Mixed and Hybrid Methods, in Handbook of Numerical Analysis, edited by P.G. Ciarlet and J.L. Lions, vol. II, Finite Element Methods (Part 1), North-Holland, Amsterdam (1991). Zbl0875.65090MR1115239
  27. [27] R. Stenberg, A family of mixed finite elements for the elasticity problem. Numer. Math.53 (1988) 513–538. Zbl0632.73063MR954768

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