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, Volume 47 (2013) no. 2, p. 471-506

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.

DOI : https://doi.org/10.1051/m2an/2012043
Classification:  65N30,  65N12,  65N15,  74F10,  74B05,  35J05
Keywords: mixed finite elements, Helmholtz equation, elastodynamic equation
@article{M2AN_2013__47_2_471_0,
author = {Dom\'\i nguez, Carolina and Gatica, Gabriel N. and Meddahi, Salim and Oyarz\'ua, Ricardo},
title = {A priori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {47},
number = {2},
year = {2013},
pages = {471-506},
doi = {10.1051/m2an/2012043},
zbl = {1267.76061},
language = {en},
url = {http://www.numdam.org/item/M2AN_2013__47_2_471_0}
}

Domínguez, Carolina; Gatica, Gabriel N.; Meddahi, Salim; Oyarzúa, Ricardo. 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, Volume 47 (2013) no. 2, pp. 471-506. doi : 10.1051/m2an/2012043. http://www.numdam.org/item/M2AN_2013__47_2_471_0/

[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. | MR 840802 | Zbl 0633.73074

[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). | MR 421106 | Zbl 0268.65052

[3] S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods. Springer-Verlag New York, Inc. (1994). | MR 1278258 | Zbl 1135.65042

[4] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Springer Verlag (1991). | MR 1115205 | Zbl 0788.73002

[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. | MR 1096235 | Zbl 0731.76043

[6] D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory. 2nd edition. Springer-Verlag, Berlin (1998). | MR 1635980 | Zbl 0760.35053

[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. | MR 2346371 | Zbl 1225.74087

[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. | MR 2376978 | Zbl 1169.74601

[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. | MR 2566767 | Zbl 1171.76027

[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. | MR 2970759 | Zbl pre06070634

[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). | MR 3218341

[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. | MR 2813344 | Zbl 1301.76047

[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). | MR 851383 | Zbl 0585.65077

[14] P. Grisvard, Elliptic Problems in Non-Smooth Domains. Pitman. Monogr. Studies Math. 24 (1985). | Zbl 0695.35060

[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. | MR 840970 | Zbl 0623.35031

[16] R. Hiptmair, Finite elements in computational electromagnetism. Acta Numer. 11 (2002) 237-339. | MR 2009375 | Zbl 1123.78320

[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. | MR 1288317 | Zbl 0849.76040

[18] G.C. Hsiao, R.E. Kleinman and G.F. Roach, Weak solutions of fluid-solid interaction problems. Math. Nachrichten 218 (2000) 139-163. | MR 1784639 | Zbl 0963.35043

[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. | MR 1000990

[20] F. Ihlenburg, Finite Element Analysis of Acoustic Scattering. Springer-Verlag, New York (1998). | MR 1639879 | Zbl 0908.65091

[21] R. Kress, Linear Integral Equ. Springer-Verlag, Berlin (1989). | MR 1007594

[22] M. Lonsing and R. Verfürth, On the stability of BDMS and PEERS elements. Numer. Math. 99 (2004) 131-140. | MR 2101787 | Zbl 1076.65090

[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. | MR 2081003 | Zbl 1127.74328

[24] W. Mclean, Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press (2000). | MR 1742312 | Zbl 0948.35001

[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. | MR 2169165 | Zbl 1078.74054

[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). | MR 1115239 | Zbl 0875.65090

[27] R. Stenberg, A family of mixed finite elements for the elasticity problem. Numer. Math. 53 (1988) 513-538. | MR 954768 | Zbl 0632.73063