Moving Dirichlet boundary conditions
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 6, p. 1859-1876
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This paper develops a framework to include Dirichlet boundary conditions on a subset of the boundary which depends on time. In this model, the boundary conditions are weakly enforced with the help of a Lagrange multiplier method. In order to avoid that the ansatz space of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, which maps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition as well as existence results are presented for a class of second order initial-boundary value problems. For the semi-discretization in space, a finite element scheme is presented which satisfies a discrete stability condition. Because of the saddle point structure of the underlying PDE, the resulting system is a DAE of index 3.

DOI : https://doi.org/10.1051/m2an/2014022
Classification:  65J10,  65M60,  65M20
Keywords: Dirichlet boundary conditions, operator DAE, inf-sup condition, wave equation
@article{M2AN_2014__48_6_1859_0,
     author = {Altmann, Robert},
     title = {Moving Dirichlet boundary conditions},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {6},
     year = {2014},
     pages = {1859-1876},
     doi = {10.1051/m2an/2014022},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_6_1859_0}
}
Altmann, Robert. Moving Dirichlet boundary conditions. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 6, pp. 1859-1876. doi : 10.1051/m2an/2014022. http://www.numdam.org/item/M2AN_2014__48_6_1859_0/

[1] R.A. Adams and J.J.F. Fournier, Sobolev Spaces, 2nd edn. Elsevier, Amsterdam (2003). | MR 2424078 | Zbl 1098.46001

[2] R. Altmann. Index reduction for operator differential-algebraic equations in elastodynamics. Z. Angew. Math. Mech. (ZAMM) 93 (2013) 648-664. | MR 3105029 | Zbl pre06221305

[3] R. Altmann. Modeling flexible multibody systems by moving Dirichlet boundary conditions. In Proc. of Multibody Dynamics 2013 - ECCOMAS Thematic Conference, Zagreb, Croatia, July 1-4 (2013).

[4] M. Arnold and B. Simeon, The simulation of pantograph and catenary: a PDAE approach. Preprint (1990), Technische Universität Darmstadt, Germany (1998).

[5] M. Arnold and B. Simeon, Pantograph and catenary dynamics: A benchmark problem and its numerical solution. Appl. Numer. Math. 34 (2000) 345-362. | MR 1782540 | Zbl 0964.65101

[6] I. Babuška, The finite element method with Lagrangian multipliers. Numer. Math. 20 (1973) 179-192. | MR 359352 | Zbl 0258.65108

[7] I. Babuška and G.N. Gatica, On the mixed finite element method with Lagrange multipliers. Numer. Meth. Part. D. E. 19 (2003) 192-210. | MR 1958060 | Zbl 1021.65056

[8] F. Ben Belgacem, The mortar finite element method with Lagrange multipliers. Numer. Math. 84 (1999) 173-197. Doi:10.1007/s002110050468. | MR 1730018 | Zbl 0944.65114

[9] D. Braess, Finite Elements - Theory, Fast Solvers, and Applications in Solid Mechanics, 3rd edn. Cambridge University Press, New York (2007). | Zbl 1118.65117

[10] J.H. Bramble, The Lagrange multiplier method for Dirichlet's problem. Math. Comput. 37 (1981) 1-11. | MR 616356 | Zbl 0477.65077

[11] S. C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, 3rd edn. Springer-Verlag, New York (2008). | MR 2373954 | Zbl 0804.65101

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

[13] F.J. Cavalieri, A. Cardona, V.D. Fachinotti and J. Risso, A finite element formulation for nonlinear 3D contact problems. Mecánica Comput. XXVI(16) (2007) 1357-1372.

[14] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978). | MR 520174 | Zbl 0511.65078

[15] E. Emmrich and D. Šiška, Evolution equations of second order with nonconvex potential and linear damping: existence via convergence of a full discretization. Technical report, University of Liverpool (2012).

[16] L.C. Evans, Partial Differential Equations, 2nd edn. American Mathematical Society (AMS). Providence (1998). | Zbl 1194.35001

[17] L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, FL (1992). | MR 1158660 | Zbl 0804.28001

[18] M. Géradin and A. Cardona, Flexible Multibody Dynamics: A Finite Element Approach. John Wiley, Chichester (2001).

[19] J.A. Griepentrog, K. Gröger, H.-C. Kaiser and J. Rehberg, Interpolation for function spaces related to mixed boundary value problems. Math. Nachr. 241 (2002) 110-120. | MR 1912381 | Zbl 1010.46021

[20] B. Gustafsson, High Order Difference Methods for Time Dependent PDE. Springer-Verlag, Berlin (2008). | MR 2380849 | Zbl 1146.65064

[21] P. Kunkel and V. Mehrmann, Differential-Algebraic Equations: Analysis and Numerical Solution. European Mathematical Society (EMS), Zürich (2006). | MR 2225970 | Zbl 1095.34004

[22] J.-L. Lions and E. Magenes, Problèmes aux Limites Non Homogènes et Applications. Vol. 1. Travaux et Recherches Mathématiques, No. 17. Dunod, Paris (1968). | Zbl 0165.10801

[23] J.-L. Lions and W.A. Strauss, Some non-linear evolution equations. Bull. Soc. Math. France 93 (1965) 43-96. | Numdam | MR 199519 | Zbl 0132.10501

[24] M.K. Lipinski, A posteriori Fehlerschätzer für Sattelpunktsformulierungen nicht-homogener Randwertprobleme. Ph.D thesis, Ruhr Universität Bochum (2004). | Zbl 1192.65143

[25] J. Nečas, Les Méthodes Directes en Théorie des Equations Elliptiques. Masson et Cie, Éditeurs, Paris (1967). | MR 227584 | Zbl 1225.35003

[26] L.E. Payne and H.F. Weinberger, An optimal Poincaré inequality for convex domains. Arch. Rational Mech. Anal. 5 (1960) 286-292. | MR 117419 | Zbl 0099.08402

[27] G. Poetsch, J. Evans, R. Meisinger, W. Kortüm, W. Baldauf, A. Veitl and J. Wallaschek, Pantograph/catenary dynamics and control. Vehicle System Dynamics 28 (1997) 159-195.

[28] A.A. Shabana, Dynamics of Multibody Systems, 3rd edn. Cambridge University Press, Cambridge (2005). | Zbl 1202.70001

[29] B. Simeon, On Lagrange multipliers in flexible multibody dynamics. Comput. Method. Appl. M 195 (2006) 6993-7005. | MR 2258325 | Zbl 1120.74517

[30] B. Simeon, Computational flexible multibody dynamics. A differential-algebraic approach. Differential-Algebraic Equations Forum. Springer-Verlag, Berlin (2013). | MR 3086702 | Zbl 1279.70002

[31] O. Steinbach, Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements. Springer-Verlag, New York (2008). | MR 2361676 | Zbl 1153.65302

[32] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner, Stuttgart (1996). | Zbl 0853.65108

[33] J. Wloka, Partial Differential Equations. Cambridge University Press, Cambridge (1987). | MR 895589 | Zbl 0623.35006

[34] E. Zeidler, Nonlinear Functional Analysis and its Applications IIa: Linear Monotone Operators. Springer-Verlag, New York (1990). | MR 1033497 | Zbl 0684.47029