Previous Articles     Next Articles

A POSTERIORI ERROR ANALYSIS OF A FULLY-MIXED FINITE ELEMENT METHOD FOR A TWO-DIMENSIONAL FLUID-SOLID INTERACTION PROBLEM

Carolina Domínguez1, Gabriel N. Gatica2, Salim Meddahi3   

  1. 1 Instituto de Ciencias Físicas y Matemáticas, Facultad de Ciencias, Universidad Austral de Chile, Avenida Eduardo Morales Miranda-Campus Isla Teja, Valdivia, Chile;
    2 CI2 MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile;
    3 Departamento de Matemáticas, Facultad de Ciencias, Universidad de Oviedo, Calvo Sotelo s/n, Oviedo, España
  • Received:2014-02-18 Revised:2015-06-05 Online:2015-11-15 Published:2015-11-15
  • Supported by:

    This research was partially supported by BASAL project CMM, Universidad de Chile, by Centro de Investigación en Ingeniería Matemática(CI2 MA), Universidad de Concepción, and by CONICYT project Anillo ACT1118(ANANUM).

Carolina Domínguez, Gabriel N. Gatica, Salim Meddahi. A POSTERIORI ERROR ANALYSIS OF A FULLY-MIXED FINITE ELEMENT METHOD FOR A TWO-DIMENSIONAL FLUID-SOLID INTERACTION PROBLEM[J]. Journal of Computational Mathematics, 2015, 33(6): 606-641.

In this paper we develop an a posteriori error analysis of a fully-mixed finite element method for a fluid-solid interaction problem in 2D. The media are governed by the elastodynamic and acoustic equations in time-harmonic regime, respectively, the transmission conditions are given by the equilibrium of forces and the equality of the corresponding normal displacements, and the fluid is supposed to occupy an annular region surrounding the solid, so that a Robin boundary condition imitating the behavior of the Sommerfeld condition is imposed on its exterior boundary. Dual-mixed approaches are applied in both domains, and the governing equations are employed 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. 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. As the main contribution of this work, we derive a reliable and efficient residual-based a posteriori error estimator for the aforedescribed coupled problem. Some numerical results confirming the properties of the estimator are also reported.

CLC Number: 

[1] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, New Jersey, 1965.

[2] A. Alonso, Error estimators for a mixed method, Numer. Math., 74(1996), 385-395.

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

[4] D.N. Arnold, F. Brezzi and J. Douglas, PEERS:A new mixed finite element method for plane elasticity, Japan J. Appl. Math., 1:2(1984), 347-367.

[5] I. Babuška and G.N. Gatica, A residual-based a posteriori error estimator for the Stokes-Darcy coupled problem, SIAM J. Numer. Anal., 48:2(2010), 498-523.

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

[7] Ph. Bouillard, Influence of the pollution on the admissible field error estimation for FE solutions of the Helmholtz equation, International Journal for Numerical Methods in Engineering, 45:7(1999), 783-800.

[8] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods, Springer Verlag, 1991.

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

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

[11] C. Carstensen and G. Dolzmann, A posteriori error estimates for mixed FEM in elasticity, Numer. Math., 81:1(1998), 187-209.

[12] P. Clément, Approximation by finite element functions using local regularisation, RAIRO Modélisation Mathématique et Analyse Numérique, 9(1975), 77-84.

[13] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, New York, Oxford, 1978.

[14] C. Domínguez, G.N. Gatica, S. Meddahi and R. Oyarzúa, A priori error analysis of a fully-mixed finite element method for a two-dimensional fluid-solid interaction problem, ESAIM:Math. Model. Numer. Anal., 47:2(2013), 471-506.

[15] G.N. Gatica, A Simple Introduction to the Mixed Finite Element Method. Theory and Applications, SpringerBriefs in Mathematics, Springer, Cham Heidelberg New York Dordrecht London, 2014.

[16] G.N. Gatica, G.C. Hsiao and S. Meddahi, A residual-based a posteriori error estimator for a two-dimensional fluid-solid interaction problem, Numerische Mathematik, 114:1(2009), 63-106.

[17] 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., 45:5(2007), 2072-2097.

[18] 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., 59:11(2009), 2735-2750,

[19] G.N. Gatica, A. Márquez and M.A. Sánchez, Analysis of a velocity-pressure-pseudostress formulation for the stationary Stokes equations, Computer Methods in Applied Mechanics and Engineering, 199:17-20(2010), 1064-1079.

[20] G.N. Gatica and S. Meddahi, An a-posteriori error estimate for the coupling of BEM and mixedFEM, Numerical Functional Analysis and Optimization, 20:5-6(1999), 449-472.

[21] 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.

[22] G.N. Gatica, R. Oyarzúa and F.-J. Sayas, A residual-based a posteriori error estimator for a fully mixed formulation of the Stokes-Darcy coupled problem, Computer Methods in Applied Mechanics and Engineering, 200:21-22(2011), 1877-1891.

[23] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, vol. 5, Springer-Verlag, 1986.

[24] F. Ihlenburg and I. Babuška, Dispersion analysis and error estimation of Galerkin finite element methods for the Helmholtz equation, International Journals for Numerical Methods in Engineering, 38:22(1995), 3745-3774.

[25] S. Irimie and Ph. Boiullard, A reliable a posteriori error estimator for the finite element solution of the Helmholtz equation, Computer Methods in Applied Mechanics and Engineering, 190:31(2001), 4027-4042.

[26] M. Lonsing and R. Verfürth, A posteriori error estimators for mixed finite element methods in linear elasticity, Numerische Mathematik, 97:4(2004), 757-778.

[27] 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), 1991, NorthHolland, Amsterdam.

[28] R. Verfürth, A posteriori error estimation and adaptive mesh-refinement techniques, Journal of Computational and Applied Mathematics, 50:1-3(1994), 67-83.

[29] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner(Chichester), 1996.
[1] Lina Dong, Shaochun Chen. UNIFORMLY CONVERGENT NONCONFORMING TETRAHEDRAL ELEMENT FOR DARCY-STOKES PROBLEM [J]. Journal of Computational Mathematics, 2019, 37(1): 130-150.
[2] Zhoufeng Wang, Peiqi Huang. AN ADAPTIVE FINITE ELEMENT METHOD FOR THE WAVE SCATTERING BY A PERIODIC CHIRAL STRUCTURE [J]. Journal of Computational Mathematics, 2018, 36(6): 845-865.
[3] Ruming Zhang, Bo Zhang. A NEW INTEGRAL EQUATION FORMULATION FOR SCATTERING BY PENETRABLE DIFFRACTION GRATINGS [J]. Journal of Computational Mathematics, 2018, 36(1): 110-127.
[4] Peijun Li. A SURVEY OF OPEN CAVITY SCATTERING PROBLEMS [J]. Journal of Computational Mathematics, 2018, 36(1): 1-16.
[5] Xiaolu Su, Xiufang Feng, Zhilin Li. FOURTH-ORDER COMPACT SCHEMES FOR HELMHOLTZ EQUATIONS WITH PIECEWISE WAVE NUMBERS IN THE POLAR COORDINATES [J]. Journal of Computational Mathematics, 2016, 34(5): 499-510.
[6] Yifeng Xu, Jianguo Huang, Xuehai Huang. A POSTERIORI ERROR ESTIMATES FOR LOCAL C0 DISCONTINUOUS GALERKIN METHODS FOR KIRCHHOFF PLATE BENDING PROBLEMS [J]. Journal of Computational Mathematics, 2014, 32(6): 665-686.
[7] Tian Luan, Fuming Ma, Minghui Liu. ERROR ESTIMATION FOR NUMERICAL METHODS USING THE ULTRA WEAK VARIATIONAL FORMULATION IN MODEL OF NEAR FIELD SCATTERING PROBLEM [J]. Journal of Computational Mathematics, 2014, 32(5): 491-506.
[8] Lunji Song, Jing Zhang, Li-Lian Wang. A MULTI-DOMAIN SPECTRAL IPDG METHOD FOR HELMHOLTZ EQUATION WITH HIGH WAVE NUMBER [J]. Journal of Computational Mathematics, 2013, 31(2): 107-136.
[9] Zhongyi Huang, Xu Yang. TAILORED FINITE CELL METHOD FOR SOLVING HELMHOLTZ EQUATION IN LAYERED HETEROGENEOUS MEDIUM [J]. Journal of Computational Mathematics, 2012, 30(4): 381-391.
[10] Yunyun Ma, Fuming Ma, Heping Dong. A PROJECTION METHOD WITH REGULARIZATION FOR THE CAUCHY PROBLEM OF THE HELMHOLTZ EQUATION [J]. Journal of Computational Mathematics, 2012, 30(2): 157-176.
[11] Xiufang Feng, Zhilin Li, Zhonghua Qiao. HIGH ORDER COMPACT FINITE DIFFERENCE SCHEMES FOR THE HELMHOLTZ EQUATION WITH DISCONTINUOUS COEFFICIENTS [J]. Journal of Computational Mathematics, 2011, 29(3): 324-340.
[12] Yanzhao Cao, Ran Zhang , Kai Zhang . Finite Element and Discontinuous Galerkin Method for Stochastic Helmholtz Equation in Two- and Three-Dimensions [J]. Journal of Computational Mathematics, 2008, 26(5): 702-715.
[13] Houde Han, Zhongyi Huang . A Tailored Finite Point Method for the Helmholtz Equation with High WaveNumbers in Heterogeneous Medium [J]. Journal of Computational Mathematics, 2008, 26(5): 728-739.
[14] Ali Yang, Liantang Wang . A Method for Solving the Inverse Scattering Problem for Shape and Impedance [J]. Journal of Computational Mathematics, 2008, 26(4): 624-632.
[15] Shangyou Zhang . On the P1 Powell-Sabin Divergence-Free Finite Element for the Stokes Equations [J]. Journal of Computational Mathematics, 2008, 26(3): 456-470.
Viewed
Full text


Abstract