Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-24T03:38:57.447Z Has data issue: false hasContentIssue false
  • English
  • Français

Variationally consistent discretization schemes and numerical algorithms for contact problems*

Published online by Cambridge University Press:  28 April 2011

Barbara Wohlmuth
Barbara Wohlmuth
Affiliation:
Technische Universität München, Fakultät für Mathematik M2, Boltzmannstr. 3, 85748 Garching, Germany E-mail: wohlmuth@ma.tum.de URL: www-m2.ma.tum.de
Get access Rights & Permissions [Opens in a new window]

Abstract

We consider variationally consistent discretization schemes for mechanical contact problems. Most of the results can also be applied to other variational inequalities, such as those for phase transition problems in porous media, for plasticity or for option pricing applications from finance. The starting point is to weakly incorporate the constraint into the setting and to reformulate the inequality in the displacement in terms of a saddle-point problem. Here, the Lagrange multiplier represents the surface forces, and the constraints are restricted to the boundary of the simulation domain. Having a uniform inf-sup bound, one can then establish optimal low-order a priori convergence rates for the discretization error in the primal and dual variables. In addition to the abstract framework of linear saddle-point theory, complementarity terms have to be taken into account. The resulting inequality system is solved by rewriting it equivalently by means of the non-linear complementarity function as a system of equations. Although it is not differentiable in the classical sense, semi-smooth Newton methods, yielding super-linear convergence rates, can be applied and easily implemented in terms of a primal–dual active set strategy. Quite often the solution of contact problems has a low regularity, and the efficiency of the approach can be improved by using adaptive refinement techniques. Different standard types, such as residual- and equilibrated-based a posteriori error estimators, can be designed based on the interpretation of the dual variable as Neumann boundary condition. For the fully dynamic setting it is of interest to apply energy-preserving time-integration schemes. However, the differential algebraic character of the system can result in high oscillations if standard methods are applied. A possible remedy is to modify the fully discretized system by a local redistribution of the mass. Numerical results in two and three dimensions illustrate the wide range of possible applications and show the performance of the space discretization scheme, non-linear solver, adaptive refinement process and time integration.

Type
Research Article
Information
Acta Numerica , Volume 20 , May 2011 , pp. 569 - 734
Copyright
Copyright © Cambridge University Press 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Achdou, Y. and Pironneau, O. (2005), Computational Methods for Option Pricing, SIAM.CrossRefGoogle Scholar
Acosta, M., Merten, C., Eigenberger, G., Class, H., Helmig, R., Thoben, B. and Müller-Steinhagen, H. (2006), ‘Modeling non-isothermal two-phase multicom-ponent flow in the cathode of PEM fuel cells’, J. Power Sour. 159, 1123–1141.CrossRefGoogle Scholar
Adams, R. (1975), Sobolev Spaces, Academic Press.Google Scholar
Ahrens, J., Geveci, B. and Law, C. (2005), ParaView: An end-user tool for large data visualization. In The Visualization Handbook (Hansen, C. D. and Johnson, C. R., eds), Elsevier, pp. 717–732. Available at: www.paraview.org.CrossRefGoogle Scholar
Ainsworth, M. and Oden, J. (1993), ‘A posteriori error estimators for 2nd order elliptic systems II: An optimal order process for calculating self-equilibrated fluxes’, Comput. Math. Appl. 26, 75–87.CrossRefGoogle Scholar
Ainsworth, M. and Oden, J. (2000), A Posteriori Error Estimation in Finite Element Analysis, Wiley.CrossRefGoogle Scholar
Ainsworth, M., Oden, J. and Lee, C. (1993), ‘Local a posteriori error estimators for variational inequalities’, Numer. Methods Partial Diff. Equations 9, 23–33.CrossRefGoogle Scholar
Alart, P. and Curnier, A. (1991), ‘A mixed formulation for frictional contact problems prone to Newton like solution methods’, Comput. Methods Appl. Mech. Engrg 92, 353–375.CrossRefGoogle Scholar
Alberty, J., Carstensen, C. and Zarrabi, D. (1999), ‘Adaptive numerical analysis in primal elastoplasticity with hardening’, Comput. Methods Appl. Mech. Engrg 171, 175–204.CrossRefGoogle Scholar
Andersen, K., Christiansen, E., Conn, A. and Overton, M. (2000), ‘An efficient primal–dual interior point method for minimizing a sum of Euclidean norms’, SIAM J. Sci. Comput. 22, 243–262.CrossRefGoogle Scholar
Armero, F. and Petöcz, E. (1998), ‘A new class of conserving algorithms for dynamic contact problems.’, Comput. Methods Appl. Mech. Engrg 158, 269–300.CrossRefGoogle Scholar
Armero, F. and Petöcz, E. (1999), ‘A new dissipative time-stepping algorithm for frictional contact problems: Formulation and analysis’, Comput. Methods Appl. Mech. Engrg 179, 151–178.CrossRefGoogle Scholar
Arnold, D. and Awanou, G. (2005), ‘Rectangular mixed finite elements for elasticity’, Math. Models Meth. Appl. Sci. 15, 1417–1429.CrossRefGoogle Scholar
Arnold, D. and Winther, R. (2002), ‘Mixed finite element methods for elasticity’, Numer. Math. 92, 401–419.CrossRefGoogle Scholar
Arnold, D. and Winther, R. (2003), Mixed finite elements for elasticity in the stress-displacement formulation. In Current Trends in Scientific Computing (Chen, Z., Glowinski, R. and Li, K., eds), Vol. 329 of Contemporary Mathematics, AMS, pp. 33–42.CrossRefGoogle Scholar
Arnold, D., Awanou, G. and Winther, R. (2008), ‘Finite elements for symmetric tensors in three dimensions’, Math. Comput. 77, 1229–1251.CrossRefGoogle Scholar
Arnold, D., Falk, R. and Winther, R. (2006), Differential complexes and stability of finite element methods II: The elasticity complex. In Compatible Spatial Discretizations (Arnold, D. N.et al., eds), Vol. 142 of The IMA Volumes in Mathematics and its Applications, Springer, pp. 47–67.CrossRefGoogle Scholar
Babuška, I. and Strouboulis, T. (2001), The Finite Element Method and its Reliability., Clarendon.CrossRefGoogle Scholar
Bajer, A. and Demkowicz, L. (2002), ‘Dynamic contact/impact problems, energy conservation, and planetary gear trains’, Comput. Methods Appl. Mech. En-gng 191, 4159–4191.CrossRefGoogle Scholar
Baker, G. and Dougalis, V. (1976), ‘The effect of quadrature errors on finite element approximations for second order hyperbolic equations’, SIAM J. Numer. Anal. 13, 577–598.CrossRefGoogle Scholar
Ballard, P. (1999), ‘A counter-example to uniqueness in quasi-static elastic contact problems with small friction’, Internat. J. Engrg Sci. 37, 163–178.CrossRefGoogle Scholar
Ballard, P. and Basseville, S. (2005), ‘Existence and uniqueness for dynamical unilateral contact with Coulomb friction: A model problem’, M2AN: Math. Model. Numer. Anal. 39, 59–77.CrossRefGoogle Scholar
Ballard, P., Léger, A. and Pratt, E. (2006), Stability of discrete systems involving shocks and friction. In Analysis and Simulation of Contact Problems (Wrig-gers, P. and Nackenhorst, U., eds), Vol. 27 of Lecture Notes in Applied and Computational Mechanics, Springer, pp. 343–350.CrossRefGoogle Scholar
Bastian, P., Birken, K., Johannsen, K., Lang, S., Neuβ, N., Rentz-Reichert, H. and Wieners, C. (1997), ‘UG: A flexible software toolbox for solving partial differential equations’, Comput. Vis. Sci. 1, 27–40.CrossRefGoogle Scholar
Bastian, P., Blatt, M., Dedner, A., Engwer, C., Klöfkorn, R., Kornhuber, R., Ohlberger, M. and Sander, O. (2008), ‘A generic grid interface for parallel and adaptive scientific computing II: Implementation and tests in DUNE’, Computing 82, 121–138.CrossRefGoogle Scholar
Bayada, G., Sabil, J. and Sassi, T. (2002), ‘Neumann–Dirichlet algorithm for unilateral contact problems: Convergence results’, CR Math. Acad. Sci. Paris 335, 381–386.CrossRefGoogle Scholar
Bayada, G., Sabil, J. and Sassi, T. (2008), ‘Convergence of a Neumann–Dirichlet algorithm for tow-body contact problems with nonlocal Coulomb's friction law’, ESAIM: Math. Model. Numer. Anal. 42, 243–262.CrossRefGoogle Scholar
Belhachmi, Z. (2003), ‘A posteriori error estimates for the 3D stabilized mortar finite element method applied to the Laplace equation’, Math. Model. Numer. Anal. 37, 991–1011.CrossRefGoogle Scholar
Belhachmi, Z. (2004), ‘Residual a posteriori error estimates for a 3D mortar finite-element method: The Stokes system’, IMA J. Numer. Anal. 24, 521–546.CrossRefGoogle Scholar
Belhachmi, Z. and Ben Belgacem, F. (2000), ‘Finite elements of order two for Signorini's variational inequality’, CR Acad. Sci. Paris, Sér. I: Math. 331, 727–732.CrossRefGoogle Scholar
Ben Belgacem, F. (2000), ‘Numerical simulation of some variational inequalities arisen from unilateral contact problems by the finite element methods’, SIAM J. Numer. Anal. 37, 1198–1216.CrossRefGoogle Scholar
Ben Belgacem, F. and Maday, Y. (1997), ‘The mortar element method for three dimensional finite elements’, M2AN: Math. Model. Numer. Anal. 31, 289– 302.CrossRefGoogle Scholar
Ben Belgacem, F. and Renard, Y. (2003), ‘Hybrid finite element methods for the Signorini problem’, Math. Comp. 72, 1117–1145.CrossRefGoogle Scholar
Ben Belgacem, F., Hild, P. and Laborde, P. (1997), ‘Approximation of the unilateral contact problem by the mortar finite element method’, CR Acad. Sci. Paris, S ér. I 324, 123–127.Google Scholar
Ben Belgacem, F., Hild, P. and Laborde, P. (1998), ‘The mortar finite element method for contact problems’, Math. Comput. Modelling 28, 263–271.CrossRefGoogle Scholar
Ben Belgacem, F., Hild, P. and Laborde, P. (1999), ‘Extension of the mortar finite element method to a variational inequality modeling unilateral contact’, Math. Models Methods Appl. Sci. 9, 287–303.CrossRefGoogle Scholar
Bergam, A., Bernardi, C., Hecht, F. and Mghazli, Z. (2003), ‘Error indicators for the mortar finite element discretization of a parabolic problem’, Numer. Algorithms 34, 187–201.CrossRefGoogle Scholar
Bernardi, C. and Hecht, F. (2002), ‘Error indicators for the mortar finite element discretization of the Laplace equation’, Math. Comput. 71, 1371–1403.CrossRefGoogle Scholar
Bernardi, C., Maday, Y. and Patera, A. (1993), Domain decomposition by the mortar element method. In Asymptotic and Numerical Methods for Partial Differential Equations With Critical Parameters (Kaper, H.et al., eds), Reidel, pp. 269–286.CrossRefGoogle Scholar
Bernardi, C., Maday, Y. and Patera, A. (1994), A new nonconforming approach to domain decomposition: The mortar element method. In Nonlinear Partial Differential Equations and their Applications (Brezis, H. and Lions, J.-L., eds), Vol. XI of Collège de France Seminar, Pitman, pp. 13–51.Google Scholar
Betsch, P. and Hesch, C. (2007), Energy-momentum conserving schemes for friction-less contact problem I: NTS method. In Computational Methods in Contact Mechanics, Vol. 3 of IUTAM, Springer, pp. 77–96.Google Scholar
Betsch, P. and Steinmann, P. (2002 a), ‘Conservation properties of a time FE method III: Mechanical systems with holonomic constraints’, Internat. J. Numer. Methods Engrg 53, 2271–2304.CrossRefGoogle Scholar
Betsch, P. and Steinmann, P. (2002 b), ‘A DAE approach to flexible multibody dynamics’, Multibody Syst. Dyn. 8, 367–391.CrossRefGoogle Scholar
Bildhauer, M., Fuchs, M. and Repin, S. (2008), ‘Duality based a posteriori error estimates for higher order variational inequalities with power growth functionals’, Ann. Acad. Sci. Fenn., Math. 33, 475–490.Google Scholar
Binev, P., Dahmen, W. and DeVore, R. (2004), ‘Adaptive finite element methods with convergence rates’, Numer. Math. 97, 219–268.CrossRefGoogle Scholar
Binning, P. and Celia, M. (1999), ‘Practical implementation of the fractional flow approach to multi-phase flow simulation’, Adv. Water Resour. 22, 461–478.CrossRefGoogle Scholar
Black, F. and Scholes, M. (1973), ‘The pricing of options and corporate liabilities’, J. Pol. Econ. 81, 637–659.CrossRefGoogle Scholar
Blum, H. and Suttmeier, F. (2000), ‘An adaptive finite element discretisation for a simplified Signorini problem’, Calcolo 37, 65–77.CrossRefGoogle Scholar
Boieri, P., Gastaldi, F. and Kinderlehrer, D. (1987), ‘Existence, uniqueness, and regularity results for the two-body contact problem’, Appl. Math. Optim. 15, 251–277.CrossRefGoogle Scholar
Borri, M., Bottasso, C. and Trainelli, L. (2001), ‘Integration of elastic multibody systems by invariant conserving/dissipating algorithms II: Numerical schemes and applications’, Comput. Methods Appl. Mech. Engrg 190, 3701–3733.CrossRefGoogle Scholar
Bostan, V. and Han, W. (2006), ‘A posteriori error analysis for finite element solutions of a frictional contact problem’, Comput. Methods Appl. Mech. Engrg 195, 1252–1274.CrossRefGoogle Scholar
Bostan, V., Han, W. and Reddy, B. (2005), ‘A posteriori error estimation and adaptive solution of elliptic variational inequalities of the second kind’, Appl. Numer. Math. 52, 13–38.CrossRefGoogle Scholar
Braess, D. (2005), ‘A posteriori error estimators for obstacle problems: Another look’, Numer. Math. 101, 523–549.CrossRefGoogle Scholar
Braess, D. and Dahmen, W. (1998), ‘Stability estimates of the mortar finite element method for 3-dimensional problems’, East–West J. Numer. Math. 6, 249–263.Google Scholar
Braess, D. and Dahmen, W. (2002), The mortar element method revisited: What are the right norms? In Domain Decomposition Methods in Science and Engineering: Thirteenth International Conference on Domain Decomposition Methods (Debit, N.et al., eds), CIMNE, pp. 27–40.Google Scholar
Braess, D., Carstensen, C. and Hoppe, R. (2007), ‘Convergence analysis of a conforming adaptive finite element method for an obstacle problem’, Numer. Math. 107, 455–471.CrossRefGoogle Scholar
Braess, D., Carstensen, C. and Hoppe, R. (2009 a), ‘Error reduction in adaptive finite element approximations of elliptic obstacle problems’, J. Comput. Math. 27, 148–169.Google Scholar
Braess, D., Carstensen, C. and Reddy, B. (2004), ‘Uniform convergence and a posteriori error estimators for the enhanced strain finite element method’, Numer. Math. 96, 461–479.Google Scholar
Braess, D., Hoppe, R. and Schöberl, J. (2008), ‘A posteriori estimators for obstacle problems by the hypercircle method’, Comput. Visual. Sci. 11, 351–362.CrossRefGoogle Scholar
Braess, D., Pillwein, V. and Schöberl, J. (2009 b), ‘Equilibrated residual error estimates are p-robust’, Comput. Methods Appl. Mech. Engrg 198, 1189–1197.CrossRefGoogle Scholar
Brandt, A. and Cryer, C. (1983), ‘Multigrid algorithms for the solution of linear complementarity problems arising from free boundary problems’, SIAM J. Sci. Statist. Comput. 4, 655–684.CrossRefGoogle Scholar
Brenan, K., Campbell, S. and Petzold, L. (1989), Numerical Solution of Initial-Value Problems in Differential-Algebraic Equations, North-Holland.Google Scholar
Brezis, H. (1971), ‘Problèmes unilatéraux’, J. Math. Pures Appl. 9, 1–168.Google Scholar
Brezzi, F. and Fortin, M. (1991), Mixed and Hybrid Finite Element Methods, Springer.CrossRefGoogle Scholar
Brezzi, F. and Marini, D. (2001), ‘Error estimates for the three-field formulation with bubble stabilization’, Math. Comput. 70, 911–934.CrossRefGoogle Scholar
Brezzi, F., Hager, W. and Raviart, P. (1977), ‘Error estimates for the finite element solution of variational inequalities’, Numer. Math. 28, 431–443.CrossRefGoogle Scholar
Brink, U. and Stein, E. (1998), ‘A posteriori error estimation in large-strain elasticity using equilibrated local Neumann problems’, Comput. Methods Appl. Mech. Engrg 161, 77–101.CrossRefGoogle Scholar
Brooks, R. and Corey, A. (1964), ‘Hydraulic properties of porous media’, Colorado State University, Fort Collins, Hydrology Paper 3, 22–27.Google Scholar
Brunβen, S. and Wohlmuth, B. (2009), ‘An overlapping domain decomposition method for the simulation of elastoplastic incremental forming processes’, Internat. J. Numer. Methods Engrg 77, 1224–1246.Google Scholar
Brunβen, S., Hager, C., Wohlmuth, B. and Schmid, F. (2008), Simulation of elasto-plastic forming processes using overlapping domain decomposition and inexact Newton methods. In IUTAM Symposium on Theoretical, Computational an d Model l ing A spects o f In e lastic Media (Reddy, B. D., ed.), Springer Science and Business media, pp. 155–164.Google Scholar
Buscaglia, G., Duran, R., Fancello, E., Feijoo, R. and Padra, C. (2001), ‘An adaptive finite element approach for frictionless contact problems’, Internat. J. Numer. Methods Engrg 50, 394–418.3.0.CO;2-#>CrossRefGoogle Scholar
Carstensen, C., Scherf, O. and Wriggers, P. (1999), ‘Adaptive finite elements for elastic bodies in contact’, SIAM J. Sci. Comput. 20, 1605–1626.CrossRefGoogle Scholar
Cascon, M., Kreuzer, C., Nochetto, R. and Siebert, K. (2008), ‘Quasi-optimal convergence rate for an adaptive finite element method’, SIAM J. Numer. Anal. 46, 2524–2550.CrossRefGoogle Scholar
Chan, T., Golub, G. and Mulet, P. (1999), ‘A nonlinear primal–dual method for total variation-based image restoration’, SIAM J. Sci. Comput. 20, 1964–1977.CrossRefGoogle Scholar
Chapelle, D. and Bathe, K. (1993), ‘The inf-sup test’, Comput. Struct. 47, 537–545.CrossRefGoogle Scholar
Chavent, G. and Jaffré, J. (1986), Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland.Google Scholar
Chawla, V. and Laursen, T. (1998), ‘Energy consistent algorithms for frictional contact problems’, Internat. J. Numer. Methods Engrg 42, 799–827.3.0.CO;2-F>CrossRefGoogle Scholar
Cheddadi, I., Fučík, R., Prieto, M. and Vohralík, M. (2008), ‘Computable a posteriori error estimates in the finite element method based on its local conservativity: Improvements using local minimization’, ESAIM: Proc. 24, 77–96.CrossRefGoogle Scholar
Cheddadi, I., Fučík, R., Prieto, M. and Vohralík, M. (2009), ‘Guaranteed and robust a posteriori error estimates for singularly perturbed reaction–diffusion problems’, ESAIM: Math. Model. Numer. Anal. 43, 867–888.CrossRefGoogle Scholar
Chen, B., Chen, X. and Kanzow, C. (2000), ‘A penalized Fischer–Burmeister NCP-function’, Math. Program., Ser. A 88, 211–216.CrossRefGoogle Scholar
Chen, J. (2007), ‘On some NCP-functions based on the generalized Fischer–Burmeister function’, Asia–Pac. J. Oper. Res. 24, 401–420.CrossRefGoogle Scholar
Chen, Z. and Nochetto, R. (2000), ‘Residual type a posteriori error estimates for elliptic obstacle problems’, Numer. Math. 84, 527–548.CrossRefGoogle Scholar
Chernov, A., Geyn, S., Maischak, M. and Stephan, E. (2006), Finite element/boundary element coupling for two-body elastoplastic contact problems with friction. In Analysis and Simulation of Contact Problems (Wriggers, P. and Nackenhorst, U., eds), Vol. 27 of Lecture Notes in Applied and Computational Mechanics, Springer, pp. 171–178.CrossRefGoogle Scholar
Chernov, A., Maischak, M. and Stephan, E. (2008), ‘hp-mortar boundary element method for two-body contact problems with friction’, Math. Meth. Appl. Sci. 31, 2029–2054.CrossRefGoogle Scholar
Christensen, P. (2002 a), ‘A nonsmooth Newton method for elastoplastic problems’, Comput. Methods Appl. Mech. Engrg 191, 1189–1219.CrossRefGoogle Scholar
Christensen, P. (2002 b), ‘A semi-smooth Newton method for elasto-plastic contact problems’, Internat. J. Solids Structures 39, 2323–2341.CrossRefGoogle Scholar
Christensen, P. and Pang, J. (1999), Frictional contact algorithms based on semis-mooth Newton methods. In Reformulation: Nonsmooth, Piecewise Smooth, Semismooth and Smoothing Methods (Fukushima, M. and Qi, L., eds), Kluwer, pp. 81–116.Google Scholar
Christensen, P., Klarbring, A., Pang, J. and Strömberg, N. (1998), ‘Formulation and comparison of algorithms for frictional contact problems’, Internat. J. Numer. Methods Engrg 42, 145–173.3.0.CO;2-L>CrossRefGoogle Scholar
Chung, J. and Hulbert, G. (1993), ‘A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized α-method’, J. Appl. Mech. 60, 371–375.CrossRefGoogle Scholar
Ciarlet, P. (1991), Basic error estimates for elliptic problems. In Finite Element Methods, Part 1 (Ciarlet, P. and Lions, J., eds), Vol. 2 of Handbook of Numerical Analysis, North-Holland, pp. 19–351.Google Scholar
Ciarlet, P. (1998), Mathematical Elasticity, Vol. I, North-Holland.Google Scholar
Class, H. (2001), Theorie und numerische Modellierung nichtisothermer Mehrphasenprozesse in NAPL-kontaminierten porösen Medien. PhD thesis, Institut für Wasserbau, Universität Stuttgart.Google Scholar
Class, H. and Helmig, R. (2002), ‘Numerical simulation of non-isothermal multiphase multicomponent processes in porous media 2: Applications for the injection of steam and air’, Adv. Water Resour. 25, 551–564.CrossRefGoogle Scholar
Class, H., Helmig, R. and Bastian, P. (2002), ‘Numerical simulation of non-isothermal multiphase multicomponent processes in porous media 1: An efficient solution technique’, Adv. Water Resour. 25, 533–550.CrossRefGoogle Scholar
Coorevits, P., Hild, P. and Pelle, J. (2000), ‘A posteriori error estimation for unilateral contact with matching and non-matching meshes’, Comput. Methods Appl. Mech. Engrg 186, 65–83.CrossRefGoogle Scholar
Coorevits, P., Hild, P., Lhalouani, K. and Sassi, T. (2001), ‘Mixed finite element methods for unilateral problems: Convergence analysis and numerical studies’, Math. Comp. 71, 1–25.CrossRefGoogle Scholar
Dautray, R. and Lions, J. (1992), Mathematical Analysis and Numerical Methods for Science and Technology: Evolution Problems, Vol. 5, Springer.Google Scholar
de Neef, M. (2000), Modelling capillary effects in heterogeneous porous media. PhD thesis, University of Delft, Netherlands.Google Scholar
De Saxcé, G. and Feng, Z. (1991), ‘New inequality and functional for contact with friction: The implicit standard material approach’, Mech. Based Des. Struct. Mach. 19, 301–325.CrossRefGoogle Scholar
Dembo, R., Eisenstat, S. and Steinhaug, T. (1982), ‘Inexact Newton methods’, SIAM J. Numer. Anal. 19, 400–408.CrossRefGoogle Scholar
Demkowicz, L. (1982), ‘On some results concerning the reciprocal formulation for the Signorini's problem’, Comput. Math. Appl. 8, 57–74.CrossRefGoogle Scholar
Demkowicz, L. and Bajer, A. (2001), ‘Conservative discretization of contact/impact problems for nearly rigid bodies’, Comput. Methods Appl. Mech. Engng 190, 1903–1924.CrossRefGoogle Scholar
Demkowicz, L. and Oden, T. (1982), ‘On some existence and uniqueness results in contact problems with nonlocal friction’, Nonlinear Anal.: Theory Methods Appl. 6, 1075–1093.CrossRefGoogle Scholar
Deuflhard, P. (2004), Newton Methods for Nonlinear Problems: Affine Invariance and Adaptive Algorithms, Springer.Google Scholar
Deuflhard, P., Krause, R. and Ertel, S. (2008), ‘A contact-stabilized Newmark method for dynamical contact problems’, Internat. J. Numer. Methods Engrg 73, 1274–1290.CrossRefGoogle Scholar
Dickopf, T. and Krause, R. (2009 a), ‘Efficient simulation of multi-body contact problems on complex geometries: A flexible decomposition approach using constrained minimization’, Internat. J. Numer. Methods Engrg 77, 1834–1862.CrossRefGoogle Scholar
Dickopf, T. and Krause, R. (2009 b), ‘Weak information transfer between non-matching warped interfaces.’, Bercovier, Michel (ed.) et al., Domain decomposition methods in science and engineering XVIII. Selected papers based on the presentations at the 18th international conference of domain decomposition methods, Jerusalem, Israel, January 12–17, 2008. Berlin: Springer. Lecture Notes in Computational Science and Engineering 70, 283–290 (2009).CrossRefGoogle Scholar
Dohrmann, C., Key, S. and Heinstein, M. (2000), ‘A method for connecting dissimilar finite element meshes in two dimensions’, Internat. J. Numer. Methods Engrg 48, 655–678.3.0.CO;2-D>CrossRefGoogle Scholar
Dörfler, W. (1996), ‘A convergent adaptive algorithm for Poisson's equation’, SIAM J. Numer. Anal. 33, 1106–1124.CrossRefGoogle Scholar
Dörsek, P. and Melenk, J. (2010), ‘Adaptive hp-FEM for the contact problem with Tresca friction in linear elasticity: The primal–dual formulation and a posteriori error estimation’, Appl. Numer. Math. 60, 689–704.CrossRefGoogle Scholar
Dostál, Z. (2009), Optimal Quadratic Programming Algorithms, with Applications to Variational Inequalities, Vol. 23 of Springer Optimization and its Applications, Springer.Google Scholar
Dostál, Z. and Horák, D. (2003), ‘Scalability and FETI based algorithm for large discretized variational inequalities’, Math. Comput. Simul. 61, 347–357.CrossRefGoogle Scholar
Dostál, Z., Friedlander, A. and Santos, S. (1998), ‘Solution of coercive and semico-ercive contact problems by FETI domain decomposition’, Contemp. Math. 218, 82–93.CrossRefGoogle Scholar
Dostál, Z., Gomes Neto, F. and Santos, S. (2000), ‘Solution of contact problems by FETI domain decomposition with natural coarse space projections’, Comput. Methods Appl. Mech. Engrg 190, 1611–1627.CrossRefGoogle Scholar
Dostál, Z., Horák, D. and Stefanica, D. (2007), ‘A scalable FETI-DP algorithm for a semi-coercive variational inequality’, Comput. Methods Appl. Mech. Engrg 196, 1369–1379.CrossRefGoogle Scholar
Dostál, Z., Horák, D. and Stefanica, D. (2009), ‘A scalable FETI-DP algorithm with non-penetration mortar conditions on contact interface’, J. Comput. Appl. Math. 231, 577–591.CrossRefGoogle Scholar
Dostál, Z., Horák, D., Kučera, R., Vondrák, V., Haslinger, J., Dobiaš, J. and Pták, S. (2005), ‘FETI based algorithms for contact problems: Scalability, large displacements and 3D Coulomb friction’, Comput. Methods Appl. Mech. Engrg 194, 395–409.CrossRefGoogle Scholar
Doyen, D. and Ern, A. (2009), ‘Convergence of a space semi-discrete modified mass method for the dynamic Signorini problem’, Commun. Math. Sci. 7, 1063– 1072.CrossRefGoogle Scholar
Duvaut, G. and Lions, J. (1976), Inequalities in Mechanics and Physics, Springer. Translation by John, C. W..CrossRefGoogle Scholar
Eck, C. (2002), ‘Existence of solutions to a thermo-viscoelastic contact problem with Coulomb friction’, Math. Models Methods Appl. Sci. 12, 1491–1511.CrossRefGoogle Scholar
Eck, C. and Jarušek, J. (1998), ‘Existence results for the static contact problem with Coulomb friction’, Math. Models Methods Appl. Sci. 8, 445–468.CrossRefGoogle Scholar
Eck, C. and Jarušek, J. (2001), ‘On the thermal aspect of dynamic contact problems’, Math. Bohem. 126, 337–352.CrossRefGoogle Scholar
Eck, C. and Jarušek, J. (2003), ‘Existence of solutions for the dynamic frictional contact problem of isotropic viscoelastic bodies’, Nonlinear Anal., Theory Methods Appl. 53, 157–181.CrossRefGoogle Scholar
Eck, C. and Wendland, W. (2003), ‘A residual-based error estimator for BEM discretizations of contact problems’, Numer. Math. 95, 253–282.CrossRefGoogle Scholar
Eck, C. and Wohlmuth, B. (2003), ‘Convergence of a contact-Neumann iteration for the solution of two-body contact problems’, Math. Models Methods Appl. Sci. 13, 1103–1118.CrossRefGoogle Scholar
Eck, C., Jarušek, J. and Krbec, M. (2005), Unilateral Contact Problems: Variational Methods and Existence Theorems, CRC Press.CrossRefGoogle Scholar
Eisenstat, S. and Walker, H. (1996), ‘Choosing the forcing terms in an inexact Newton method’, SIAM J. Sci. Comput. 17, 16–32.CrossRefGoogle Scholar
Erdmann, B., Frei, M., Hoppe, R., Kornhuber, R. and Wiest, U. (1993), ‘Adaptive finite element methods for variational inequalities’, East–West J. Numer. Math. 1, 165–197.Google Scholar
Ern, A. and Vohralík, M. (2009), ‘Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids’, CR Math. Acad. Sci. Paris 347, 441–444.CrossRefGoogle Scholar
Evans, L. (1998), Partial Differential Equations, AMS.Google Scholar
Facchinei, F. and Pang, J. (2003 a), Finite-Dimensional Variational Inequalities and Complementary Problems, Vol. I, Springer Series in Operations Research.Google Scholar
Facchinei, F. and Pang, J. (2003 b), Finite-Dimensional Variational Inequalities and Complementary Problems, Vol. II, Springer Series in Operations Research.Google Scholar
Falk, R. (1974), ‘Error estimates for the approximation of a class of variational inequalities’, Math. Comp. 28, 963–971.CrossRefGoogle Scholar
Felippa, C. (2000), ‘On the original publication of the general canonical functional of linear elasticity’, J. Appl. Mech. 67, 217–219.CrossRefGoogle Scholar
Fichera, G. (1964), ‘Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno’, Mem. Accad. Naz. Lincei 8, 91–140.Google Scholar
Fischer, A. (1992), ‘A special Newton-type optimization method’, Optimization 24, 269–284.CrossRefGoogle Scholar
Fischer-Cripps, A. (2000), Introduction to Contact Mechanics, Springer Mechanical Engineering Series.Google Scholar
Fischer, K. and Wriggers, P. (2006), ‘Mortar based frictional contact formulation for higher order interpolations using the moving friction cone’, Comput. Methods Appl. Mech. Engrg 195, 5020–5036.CrossRefGoogle Scholar
Flemisch, B., Fritz, J., Helmig, R., Niessner, J. and Wohlmuth, B. (2007), DUMUX: A multi-scale multi-physics toolbox for flow and transport processes in porous media. In ECCOMAS Thematic Conference on Multi-Scale Computational Methods for Solids and Fluids (Ibrahimbegovic, A. and Dias, F., eds), Cachan, France, pp. 82–87.Google Scholar
Flemisch, B., Melenk, J. and Wohlmuth, B. (2005 a), ‘Mortar methods with curved interfaces’, Appl. Numer. Math. 54, 339–361.CrossRefGoogle Scholar
Flemisch, B., Puso, M. and Wohlmuth, B. (2005 b), ‘A new dual mortar method for curved interfaces: 2D elasticity’, Internat. J. Numer. Methods Engrg 63, 813–832.CrossRefGoogle Scholar
Fluegge, S., ed. (1972), Handbuch der Physik, Vol. VIa, chapter on Linear Ther-moelasticity, Springer, pp. 297–346.Google Scholar
French, D., Larsson, S. and Nochetto, R. (2001), ‘Pointwise a posteriori error analysis for an adaptive penalty finite element method for the obstacle problem’, Comput. Methods Appl. Math. 1, 18–38.CrossRefGoogle Scholar
Fuchs, M. and Repin, S. (2010), ‘Estimates of the deviations from the exact solutions for variational inequalities describing the stationary flow of certain viscous incompressible fluids’, Math. Methods Appl. Sci. 33, 1136–1147.CrossRefGoogle Scholar
Geiger, C. and Kanzow, C. (2002), Theorie und Numerik Restringierter Optimierungsaufgaben, Springer.CrossRefGoogle Scholar
Gitterle, M., Popp, A., Gee, M. and Wall, W. (2010), ‘Finite deformation frictional mortar contact using a semi-smooth Newton method with consistent linearization’, Internat. J. Numer. Methods Engrg 84, 543–571.CrossRefGoogle Scholar
Glowinski, R. (1984), Numerical Methods for Nonlinear Variational Problems, Springer.CrossRefGoogle Scholar
Glowinski, R. and Le Tallec, P. (1989), Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics, Vol. 9 of SIAM Studies in Applied Mathematics.CrossRefGoogle Scholar
Glowinski, R., Lions, J. and Trémolières, R. (1981), Numerical Analysis of Variational Inequalities, North-Holland.Google Scholar
Gonzales, M., Schmidt, B. and Ortiz, M. (2010), ‘Energy-stepping integrators in Lagrangian mechanics’, Internat. J. Numer. Methods Engrg 82, 205–241.CrossRefGoogle Scholar
Gonzalez, O. (2000), ‘Exact energy and momentum conserving algorithms for general models in nonlinear elasticity’, Comput. Methods Appl. Mech. Engrg 190, 1763–1783.CrossRefGoogle Scholar
Gordon, W. and Hall, C. (1973 a), ‘Construction of curvilinear co-ordinate systems and applications to mesh generation’, Internat. J. Numer. Methods Engng 7, 461–477.CrossRefGoogle Scholar
Gordon, W. and Hall, C. (1973 b), ‘Transfinite element methods: Blending-function interpolation over arbitrary curved element domains’, Numer. Math. 21, 109– 129.CrossRefGoogle Scholar
Gwinner, J. (2009), ‘On the p-version approximation in the boundary element method for a variational inequality of the second kind modelling unilateral contact and given friction’, Appl. Numer. Math. 59, 2774–2784.CrossRefGoogle Scholar
Hackbusch, W. (1985), Multi-Grid Methods and Applications, Springer.CrossRefGoogle Scholar
Hackbusch, W. and Mittelmann, H. (1983), ‘On multigrid methods for variational inequalities’, Numer. Math. 42, 65–76.CrossRefGoogle Scholar
Hager, C. (2010), Robust numerical algorithms for dynamic frictional contact problems with different time and space scales. PhD thesis, IANS, Universität Stuttgart.Google Scholar
Hager, C. and Wohlmuth, B. (2009 a), ‘Analysis of a space-time discretization for dynamic elasticity problems based on mass-free surface elements’, SIAM J. Numer. Anal. 47, 1863–1885.CrossRefGoogle Scholar
Hager, C. and Wohlmuth, B. (2009 b), ‘Nonlinear complementarity functions for plasticity problems with frictional contact’, Comput. Methods Appl. Mech. Engrg 198, 3411–3427.CrossRefGoogle Scholar
Hager, C. and Wohlmuth, B. (2010), ‘Semismooth Newton methods for variational problems with inequality constraints’, GAMM–Mitt. 33, 8–24.CrossRefGoogle Scholar
Hager, C., Hauret, P., Le Tallec, P. and Wohlmuth, B. (2010 a), Overlapping domain decomposition for multiscale dynamic contact problems. Technical report IANS Preprint 2010/007, Universität Stuttgart.Google Scholar
Hager, C., Huëber, S. and Wohlmuth, B. (2008), ‘A stable energy conserving approach for frictional contact problems based on quadrature formulas’, Internat. J. Numer. Methods Engrg 73, 205–225.CrossRefGoogle Scholar
Hager, C., Huëber, S. and Wohlmuth, B. (2010 b), ‘Numerical techniques for the valuation of basket options and its Greeks’, J. Comput. Fin. 13, 1–31.Google Scholar
Hairer, E. and Wanner, G. (1991), Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer.CrossRefGoogle Scholar
Han, W. (2005), A Posteriori Error Analysis via Duality Theory: With Applications in Modeling and Numerical Approximations, Springer.Google Scholar
Han, W. and Reddy, B. (1995), ‘Computational plasticity: The variational basis and numerical analysis’, Comput. Mech. Advances 2, 283–400.Google Scholar
Han, W. and Reddy, B. (1999), Plasticity: Mathematical Theory and Numerical Analysis, Springer.Google Scholar
Han, W. and Sofonea, M. (2000), ‘Numerical analysis of a frictionless contact problem for elastic-viscoplastic materials’, Comput. Methods Appl. Mech. Engrg 190, 179–191.CrossRefGoogle Scholar
Han, W. and Sofonea, M. (2002), Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity, Studies in Advanced Mathematics, AMS, International Press.CrossRefGoogle Scholar
Harker, P. and Pang, J. (1990), ‘Finite-dimensional variational inequality and nonlinear complementarity problems: A survey of theory, algorithms and applications’, Math. Progr. 48, 161–220.CrossRefGoogle Scholar
Hartmann, S., Brunβen, S., Ramm, E. and Wohlmuth, B. (2007), ‘Unilateral nonlinear dynamic contact of thin-walled structures using a primal–dual active set strategy’, Internat. J. Numer. Meth. Engrg 70, 883–912.CrossRefGoogle Scholar
Haslinger, J. and Hlaváček, I. (1981), ‘Contact between two elastic bodies II: Finite element analysis’, Aplikace Mathematiky 26, 263–290.Google Scholar
Haslinger, J., Hlaváček, I. and Nečas, J. (1996), Numerical methods for unilateral problems in solid mechanics. In Handbook of Numerical Analysis (Ciarlet, P. and Lions, J.-L., eds), Vol. IV, North-Holland, pp. 313–485.Google Scholar
Haslinger, J., Hlaváček, I., Nečas, J. and Lovíšsek, J. (1988), Solution of Variational Inequalities in Mechanics, Springer.Google Scholar
Hassanizadeh, S. and Gray, W. (1993), ‘Thermodynamic basis of capillary pressure in porous media’, Water Resour. Research 29, 3389–3405.CrossRefGoogle Scholar
Hassanizadeh, S., Celia, M. and Dahle, H. (2002), ‘Experimental measurements of saturation overshoot on infiltration’, Vadose Zone J. 1, 38–57.CrossRefGoogle Scholar
Hauret, P. and Le Tallec, P. (2006), ‘Energy-controlling time integration methods for nonlinear elastodynamics and low-velocity impact’, Comput. Methods Appl. Mech. Engrg 195, 4890–4916.CrossRefGoogle Scholar
Hauret, P. and Le Tallec, P. (2007), ‘A discontinuous stabilized mortar method for general 3D elastic problems’, Comput. Methods Appl. Mech. Engng 196, 4881–4900.CrossRefGoogle Scholar
Hauret, P., Salomon, J., Weiss, A. and Wohlmuth, B. (2008), ‘Energy consistent co-rotational schemes for frictional contact problems’, SIAM J. Sci. Comput. 30, 2488–2511.CrossRefGoogle Scholar
Helmig, R. (1997), Multiphase Flow and Transport Processes in the Subsurface, Springer.CrossRefGoogle Scholar
Helmig, R., Weiss, A. and Wohlmuth, B. (2009), ‘Variational inequalities for modeling flow in heterogeneous porous media with entry pressure’, Comput. Geosci. 13, 373–390.CrossRefGoogle Scholar
Hertz, H. (1882), ‘Über die Berührung fester elastischer Körper’, J. Reine Angew. Math. 92, 156–171.CrossRefGoogle Scholar
Hesch, C. and Betsch, P. (2006), ‘A comparison of computational methods for large deformation contact problems of flexible bodies’, ZAMM: Z. Angew. Math. Mech. 86, 818–827.CrossRefGoogle Scholar
Hesch, C. and Betsch, P. (2009), ‘A mortar method for energy-momentum conserving schemes in frictionless dynamic contact problems’, Internat. J. Numer. Methods Engrg 77, 1468–1500.CrossRefGoogle Scholar
Hesch, C. and Betsch, P. (2010), ‘Transient three-dimensional domain decomposition problems: Frame-indifferent mortar constraints and conserving integration’, Internat. J. Numer. Methods Engrg 82, 329–358.CrossRefGoogle Scholar
Hilber, H., Hughes, T. and Taylor, R. (1977), ‘Improved numerical dissipation for time integration algorithms in structural dynamics’, Earthquake Engrg Struct. Dyn. 5, 283–292.CrossRefGoogle Scholar
Hild, P. (2000), ‘Numerical implementation of two nonconforming finite element methods for unilateral contact’, Comput. Methods Appl. Mech. Engrg 184, 99–123.CrossRefGoogle Scholar
Hild, P. (2003), ‘An example of nonuniqueness for the continuous static unilateral contact model with Coulomb friction’, CR Math. Acad. Sci. Paris 337, 685– 688.CrossRefGoogle Scholar
Hild, P. (2004), ‘Non-unique slipping in the Coulomb friction model in two-dimensional linear elasticity’, Q. J. Mech. Appl. Math. 57, 225–235.CrossRefGoogle Scholar
Hild, P. and Laborde, P. (2002), ‘Quadratic finite element methods for unilateral contact problems’, Appl. Numer. Math. 41, 410–421.CrossRefGoogle Scholar
Hild, P. and Lleras, V. (2009), ‘Residual error estimators for Coulomb friction’, SIAM J. Numer. Anal. 47, 3550–3583.CrossRefGoogle Scholar
Hild, P. and Nicaise, S. (2005), ‘A posteriori error estimations of residual type for Signorini's problem’, Numer. Math. 101, 523–549.CrossRefGoogle Scholar
Hild, P. and Nicaise, S. (2007), ‘Residual a posteriori error estimators for contact problems in elasticity’, Math. Model. Numer. Anal. 41, 897–923.CrossRefGoogle Scholar
Hild, P. and Renard, Y. (2006), Local uniqueness results for the discrete friction problem. In Analysis and Simulation of Contact Problems (Wriggers, P. and Nackenhorst, U., eds), Vol. 27 of Lecture Notes in Applied and Computational Mechanics, Springer, pp. 129–136.CrossRefGoogle Scholar
Hild, P. and Renard, Y. (2007), ‘An error estimate for the Signorini problem with Coulomb friction approximated by finite elements’, SIAM J. Numer. Anal. 45, 2012–2031.CrossRefGoogle Scholar
Hild, P. and Renard, Y. (2010), ‘A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics’, Numer. Math. 115, 101–129.CrossRefGoogle Scholar
Hintermüller, M. and Stadler, G. (2006), ‘An infeasible primal–dual algorithm for total variation-based inf-convolution-type image restoration’, SIAM J. Sci. Comput. 28, 1–23.CrossRefGoogle Scholar
Hintermüller, M., Ito, K. and Kunisch, K. (2002), ‘The primal–dual active set strategy as a semi–smooth Newton method’, SIAM J. Optim. 13, 865–888.CrossRefGoogle Scholar
Hintermüller, M., Kovtunenko, V. and Kunisch, K. (2004), ‘Semismooth Newton methods for a class of unilaterally constrained variational problems’, Adv. Math. Sci. Appl. 14, 513–535.Google Scholar
Hoppe, R. (1987), ‘Multigrid algorithms for variational inequalities’, SIAM J. Numer. Anal. 24, 1046–1065.CrossRefGoogle Scholar
Hoppe, R. and Kornhuber, R. (1994), ‘Adaptive multilevel methods for obstacle problems’, SIAM J. Numer. Anal. 31, 301–323.CrossRefGoogle Scholar
Hu, H. (1955), ‘On some variational principles in the theory of elasticity and the theory of plasticity’, Scientia Sinica 4, 33–54.Google Scholar
Hu, S., Huang, Z. and Chen, J. (2009), ‘Properties of a family of generalized NCP-functions and a derivative free algorithm for complementarity problems’, J. Comput. Appl. Math. 230, 69–82.CrossRefGoogle Scholar
Huber, R. and Helmig, R. (2000), ‘Node-centered finite volume discretizations for the numerical simulation of multiphase flow in heterogeneous porous media’, Comput. Geosci. 4, 141–164.CrossRefGoogle Scholar
Huëber, S. (2008), Discretization techniques and efficient algorithms for contact problems. PhD thesis, IANS, Universität Stuttgart.Google Scholar
Huëber, S. and Wohlmuth, B. (2005 a), ‘An optimal a priori error estimate for nonlinear multibody contact problems’, SIAM J. Numer. Anal. 43, 157–173.CrossRefGoogle Scholar
Huëber, S. and Wohlmuth, B. (2005 b), ‘A primal–dual active set strategy for nonlinear multibody contact problems’, Comput. Methods Appl. Mech. Engrg 194, 3147–3166.CrossRefGoogle Scholar
Huëber, S. and Wohlmuth, B. (2009), ‘Thermo-mechanical contact problem on non-matching meshes’, Comput. Methods Appl. Mech. Engrg 198, 1338–1350.CrossRefGoogle Scholar
Huëber, S. and Wohlmuth, B. (2010), Equilibration techniques for solving contact problems with Coulomb friction. Comput. Methods Appl. Mech. Engrg doi:10.1016/j.cma.2010.12.021.CrossRefGoogle Scholar
Huëber, S., Mair, M. and Wohlmuth, B. (2005 a), ‘A priori error estimates and an inexact primal–dual active set strategy for linear and quadratic finite elements applied to multibody contact problems’, Appl. Numer. Math. 54, 555–576.CrossRefGoogle Scholar
Huëber, S., Matei, A. and Wohlmuth, B. (2005 b), ‘A mixed variational formulation and an optimal a priori error estimate for a frictional contact problem in elasto-piezoelectricity’, Bull. Math. Soc. Sci. Math. Roumanie 48, 209–232.Google Scholar
Huëber, S., Matei, A. and Wohlmuth, B. (2007), ‘Efficient algorithms for problems with friction’, SIAM J. Sci. Comput. 29, 70–92.CrossRefGoogle Scholar
Huëber, S., Stadler, G. and Wohlmuth, B. (2008), ‘A primal–dual active set algorithm for three-dimensional contact problems with Coulomb friction’, SIAM J. Sci. Comput. 30, 572–596.CrossRefGoogle Scholar
Hughes, T. (1987), The Finite Element Method: Linear, Static and Dynamic Finite Element Analysis, Prentice-Hall.Google Scholar
Hulbert, G. (1992), ‘Time finite element methods for structural dynamics’, Internat. J. Numer. Methods Engrg 33, 307–331.CrossRefGoogle Scholar
Hull, J. (2006), Options, Futures, and Other Derivatives, sixth edition, Prentice-Hall.Google Scholar
Ito, K. and Kunisch, K. (2003), ‘Semi-smooth Newton methods for variational inequalities of the first kind’, M2AN: Math. Model. Numer. Anal. 37, 41–62.CrossRefGoogle Scholar
Ito, K. and Kunisch, K. (2004), ‘The primal–dual active set method for nonlinear optimal control problems with bilateral constraints’, SIAM J. Control. Optim. 43, 357–376.CrossRefGoogle Scholar
Ito, K. and Kunisch, K. (2008 a), Lagrange Multiplier Approach to Variational Problems and Applications, SIAM.CrossRefGoogle Scholar
Ito, K. and Kunisch, K. (2008 b), ‘On a semi-smooth Newton method for the Signorini problem’, Appl. Math. 53, 455–468.CrossRefGoogle Scholar
Jarušek, J. (1983), ‘Contact problems with bounded friction: Coercive case’, Czech. Math. J. 33, 237–261.CrossRefGoogle Scholar
Johnson, C. (1992), ‘Adaptive finite element methods for the obstacle problem’, Math. Models Methods Appl. Sci. 2, 483–487.CrossRefGoogle Scholar
Johnson, K. (1985), Contact Mechanics, Cambridge University Press.CrossRefGoogle Scholar
Kane, C., Marsden, J., Ortiz, M. and West, M. (2000), ‘Variational integrators and the Newmark algorithm for conservative and dissipative mechanical systems’, Internat. J. Numer. Methods Engrg 49, 1295–1325.3.0.CO;2-W>CrossRefGoogle Scholar
Kanzow, C., Yamashita, N. and Fukushima, M. (1997), ‘New NCP-functions and their properties’, J. Optimization Theory Appl. 94, 115–135.CrossRefGoogle Scholar
Karypis, G. and Kumar, V. (1998), ‘A fast and high quality multilevel scheme for partitioning irregular graphs’, SIAM J. Sci. Comput. 20, 359–392.CrossRefGoogle Scholar
Kasper, E. and Taylor, R. (2000 a), ‘A mixed-enhanced strain method I: Geometrically linear problems’, Computers and Structures 75, 237–250.CrossRefGoogle Scholar
Kasper, E. and Taylor, R. (2000 b), ‘A mixed-enhanced strain method II: Geometrically nonlinear problems’, Computers and Structures 75, 251–260.CrossRefGoogle Scholar
Kelly, D. (1984), ‘The self-equilibration of residuals and complementary a posteriori error estimates in the finite element method’, Internat. J. Numer. Methods Engrg 20, 1491–1506.CrossRefGoogle Scholar
Kelly, D. and Isles, J. (1989), ‘Procedures for residual equilibration and local error estimation in the finite element method’, Commun. Appl. Numer. Methods 5, 497–505.CrossRefGoogle Scholar
Khenous, H., Laborde, P. and Renard, Y. (2006 a), ‘Comparison of two approaches for the discretization of elastodynamic contact problems’, CR Math. Acad. Sci. Paris 342, 791–796.CrossRefGoogle Scholar
Khenous, H., Laborde, P. and Renard, Y. (2006 b), On the discretization of contact problems in elastodynamics. In Analysis and Simulation of Contact Problems (Wriggers, P. and Nackenhorst, U., eds), Vol. 27 of Lecture Notes in Applied and Computational Mechanics, Springer, pp. 31–38.CrossRefGoogle Scholar
Khenous, H., Laborde, P. and Renard, Y. (2008), ‘Mass redistribution method for finite element contact problems in elastodynamics’, Eur. J. Mech., A, Solids 27, 918–932.CrossRefGoogle Scholar
Kikuchi, N. and Oden, J. (1988), Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods, Vol. 8 of SIAM Studies in Applied Mathematics.CrossRefGoogle Scholar
Kinderlehrer, D. and Stampacchia, G. (2000), An Introduction to Variational Inequalities and their Applications, SIAM.CrossRefGoogle Scholar
Klapproth, C., Deuflhard, P. and Schiela, A. (2009), ‘A perturbation result for dynamical contact problems’, Numer. Math., Theory Methods Appl. 2, 237–257.Google Scholar
Klapproth, C., Schiela, A. and Deuflhard, P. (2010), ‘Consistency results on Newmark methods for dynamical contact problems’, Numer. Math. 116, 65–94.CrossRefGoogle Scholar
Kornhuber, R. (1994), ‘Monotone multigrid methods for elliptic variational inequalities I’, Numer. Math. 69, 167–184.CrossRefGoogle Scholar
Kornhuber, R. (1996), ‘Monotone multigrid methods for elliptic variational inequalities II’, Numer. Math. 72, 481–499.CrossRefGoogle Scholar
Kornhuber, R. (1997), Adaptive Monotone Multigrid Methods for Nonlinear Variational Problems, Teubner.Google Scholar
Kornhuber, R. and Krause, R. (2001), ‘Adaptive multigrid methods for Signorini's problem in linear elasticity’, Comput. Vis. Sci. 4, 9–20.CrossRefGoogle Scholar
Kornhuber, R. and Zou, Q. (2011), ‘Efficient and reliable hierarchical error estimates for the discretization error of elliptic obstacle problems’, Math. Comp. 80, 69– 88.CrossRefGoogle Scholar
Kornhuber, R., Krause, R., Sander, O., Deuflhard, P. and Ertel, S. (2007), ‘A monotone multigrid solver for two body contact problems in biomechanics’, Comput. Vis. Sci. 11, 3–15.CrossRefGoogle Scholar
Koziara, T. and Bicanic, N. (2008), ‘Semismooth Newton method for frictional contact between pseudo-rigid bodies’, Comput. Methods Appl. Mech. Engrg 197, 2763–2777.CrossRefGoogle Scholar
Krause, R. (2008), On the multiscale solution of constrained minimization problems. In Domain Decomposition Methods in Science and Engineering XVII (Langer, U.et al. eds), Vol. 60 of Lecture Notes in Computational Science and Engineering, Springer, pp. 93–104.CrossRefGoogle Scholar
Krause, R. (2009), ‘A nonsmooth multiscale method for solving frictional two-body contact problems in 2D and 3D with multigrid efficiency’, SIAM J. Sci. Comput. 31, 1399–1423.CrossRefGoogle Scholar
Krause, R. and Mohr, C. (2011), ‘Level set based multi-scale methods for large deformation contact problems’, Appl. Numer. Math. 61, 428–442.CrossRefGoogle Scholar
Krause, R. and Walloth, M. (2009), ‘A time discretization scheme based on Rothe's method for dynamical contact problems with friction’, Comput. Methods Appl. Mech. Engrg 199, 1–19.CrossRefGoogle Scholar
Krause, R. and Wohlmuth, B. (2002), ‘A Dirichlet–Neumann type algorithm for contact problems with friction’, Comput. Vis. Sci. 5, 139–148.CrossRefGoogle Scholar
Kuczma, M. and Demkowicz, L. (1992), ‘An adaptive algorithm for unilateral visco-elastic contact problems for beams and plates’, Comput. Methods Appl. Mech. Engng 101, 183–196.CrossRefGoogle Scholar
Kuhl, D. and Ramm, E. (1999), ‘Generalized energy-momentum method for nonlinear adaptive shell dynamics’, Comput. Methods Appl. Mech. Engrg pp. 343–366.CrossRefGoogle Scholar
Lacour, C. and Ben Belgacem, F. (2011), The Mortar Finite Element Method: Basics, Theory and Implementation, Chapman & Hall/CRC Press. To appear.Google Scholar
Ladevèze, P. and Leguillon, D. (1983), ‘Error estimate procedure in the finite element method and applications’, SIAM J. Numer. Anal. 20, 485–509.CrossRefGoogle Scholar
Ladevèze, P. and Maunder, E. (1996), ‘A general method for recovering equilibrating element tractions’, Comput. Methods Appl. Mech. Engrg 137, 111–151.CrossRefGoogle Scholar
Ladevèze, P. and Rougeot, P. (1997), ‘New advances on a posteriori error on constitutive relation in f.e. analysis’, Comput. Methods Appl. Mech. Engrg 150, 239–249.CrossRefGoogle Scholar
Lamichhane, B. and Wohlmuth, B. (2007), ‘Biorthogonal bases with local support and approximation properties’, Math. Comp. 76, 233–249.CrossRefGoogle Scholar
Lamichhane, B., Reddy, B. and Wohlmuth, B. (2006), ‘Convergence in the incompressible limit of finite element approximations based on the Hu–Washizu formulation’, Numer. Math. 104, 151–175.CrossRefGoogle Scholar
Laursen, T. (2002), Computational Contact and Impact Mechanics, Springer.Google Scholar
Laursen, T. and Chawla, V. (1997), ‘Design of energy conserving algorithms for frictionless dynamic contact problems’, Internat. J. Numer. Methods Engrg 40, 836–886.3.0.CO;2-V>CrossRefGoogle Scholar
Laursen, T. and Love, G. (2002), ‘Improved implicit integrators for transient impact problems: Geometric admissibility within the conserving framework’, Internat. J. Numer. Methods Engrg 53, 245–274.CrossRefGoogle Scholar
Laursen, T. and Meng, X. (2001), ‘A new solution procedure for application of energy-conserving algorithms to general constitutive models in nonlinear elas-todynamics’, Comput. Methods Appl. Mech. Engrg 190, 6309–6322.CrossRefGoogle Scholar
Laursen, T. and Simo, J. (1993 a), ‘A continuum-based finite element formulation for the implicit solution of multibody, large deformation frictional contact problems’, Internat. J. Numer. Methods Engrg 36, 3451–3485.CrossRefGoogle Scholar
Laursen, T. and Simo, J. (1993 b), ‘Algorithmic symmetrization of Coulomb frictional problems using augmented Lagrangians’, Comput. Methods Appl. Mech. Engng 108, 133–146.CrossRefGoogle Scholar
Lauser, A., Hager, C., Helmig, R. and Wohlmuth, B. (2010), A new approach for phase transitions in miscible multi-phase flow in porous media. SimTech-Preprint 2010–34, Universität Stuttgart. To appear In Adv. Water Resour.Google Scholar
Lee, C. and Oden, J. (1994), ‘A posteriori error estimation of h-p finite element approximations of frictional contact problems.’, Comput. Methods Appl. Mech. Engrg 113, 11–45.CrossRefGoogle Scholar
Lenhard, R., Parker, J. and Mishra, S. (1989), ‘On the correspondence between Brooks–Corey and Van Genuchten models’, J. Irrig. and Drain. Engrg 115, 744–751.CrossRefGoogle Scholar
Leverett, M. (1941), ‘Capillary behavior in porous solids’, AIME Petroleum Transactions 142, 152–169.CrossRefGoogle Scholar
Lhalouani, K. and Sassi, T. (1999), ‘Nonconforming mixed variational inequalities and domain decomposition for unilateral problems’, East–West J. Numer. Math. 7, 23–30.Google Scholar
Li, J., Melenk, J., Wohlmuth, B. and Zou, J. (2010), ‘Optimal a priori estimates for higher order finite elements for elliptic interface problems’, Appl. Numer. Math. 60, 19–37.CrossRefGoogle Scholar
Lions, J. and Stampacchia, G. (1967), ‘Variational inequalities’, Comm. Pure Appl. Math. XX, 493–519.CrossRefGoogle Scholar
Liu, W. and Yan, N. (2000), ‘A posteriori error estimators for a class of variational inequalities’, J. Sci. Comput. 15, 361–393.CrossRefGoogle Scholar
Luce, R. and Wohlmuth, B. (2004), ‘A local a posteriori error estimator based on equilibrated fluxes’, SIAM J. Numer. Anal. 42, 1394–1414.CrossRefGoogle Scholar
Lunk, C. and Simeon, B. (2006), ‘Solving constrained mechanical systems by the family of Newmark and α-methods’, Z. Angew. Math. Mech. 86, 772–784.CrossRefGoogle Scholar
Maischak, M. and Stephan, E. (2005), ‘Adaptive hp-versions of BEM for Signorini problems’, Appl. Numer. Math. 54, 425–449.CrossRefGoogle Scholar
Maischak, M. and Stephan, E. (2007), ‘Adaptive hp-versions of boundary element methods for elastic contact problems’, Comput. Mech. 39, 597–607.CrossRefGoogle Scholar
Martins, J., Barbarin, S., Raous, M. and Pinto da Costa, A. (1999), ‘Dynamic stability of finite dimensional linearly elastic systems with unilateral contact and Coulomb friction’, Comput. Methods Appl. Mech. Engrg 177, 289–328.CrossRefGoogle Scholar
Melenk, M. and Wohlmuth, B. (2011), On the convergence of surface based Lagrange multipliers in finite element methods. In preparation.Google Scholar
Moon, K., Nochetto, R., von Petersdorff, T. and Zhang, C. (2007), ‘A posteriori error analysis for parabolic variational inequalities’, ESAIM: Math. Model. Numer. Anal. 41, 485–511.CrossRefGoogle Scholar
Moreau, J. (1977), ‘Evolution problem associated with a moving convex set in a Hilbert space’, J. Differential Equations 26, 347–374.CrossRefGoogle Scholar
Morin, P., Nochetto, R. and Siebert, K. (2002), ‘Convergence of adaptive finite element methods’, SIAM Rev. 44, 631–658.CrossRefGoogle Scholar
Nečas, J., Jarušek, J. and Haslinger, J. (1980), ‘On the solution of the variational inequality to the Signorini problem with small friction’, Boll. Unione Mat. Ital., V. Ser., B 17, 796–811.Google Scholar
Nicaise, S., Witowski, K. and Wohlmuth, B. (2008), ‘An a posteriori error estimator for the Lame equation based on H(div)-conforming stress approximations’, IMA J. Numer. Anal. 28, 331–353.CrossRefGoogle Scholar
Nicolaides, R. (1982), ‘Existence, uniqueness and approximation for generalized saddle point problems’, SIAM J. Numer. Anal. 19, 349–357.CrossRefGoogle Scholar
Niessner, J. and Helmig, R. (2007), ‘Multi-scale modeling of three-phase-three-component processes in heterogeneous porous media’, Adv. Water Resour. 30, 2309–2325.CrossRefGoogle Scholar
Nochetto, R. and Wahlbin, L. (2002), ‘Positivity preserving finite element approximation’, Math. Comput. 71, 1405–1419.CrossRefGoogle Scholar
Nochetto, R., von Petersdorff, T. and Zhang, C. (2010), ‘A posteriori error analysis for a class of integral equations and variational inequalities’, Numer. Math. 116, 519–552.CrossRefGoogle Scholar
Nochetto, R., Siebert, K. and Veeser, A. (2003), ‘Pointwise a posteriori error control for elliptic obstacle problems’, Numer. Math. 95, 163–195.CrossRefGoogle Scholar
Nochetto, R., Siebert, K. and Veeser, A. (2005), ‘Fully localized a posteriori error estimators and barrier sets for contact problems’, SIAM J. Numer. Anal. 42, 2118–2135.CrossRefGoogle Scholar
Nochetto, R., Siebert, K. and Veeser, A. (2009), Theory of adaptive finite element methods: An introduction. In Multiscale, Nonlinear and Adaptive Approximation: Dedicated to Wolfgang Dahmen on the Occasion of his 60th Birthday (DeVore, R.et al., eds), Springer, pp. 409–542.CrossRefGoogle Scholar
Oden, T., Becker, E., Lin, T. and Demkowicz, L. (1985), Formulation and finite element analysis of a general class of rolling contact problems with finite elastic deformations. In The Mathematics of Finite Elements and Applications V: MAFELAP 1984, pp. 505–532.CrossRefGoogle Scholar
Pandolfi, A., Kane, C., Marsden, J. and Ortiz, M. (2002), ‘Time-discretized variational formulation of non-smooth frictional contact’, Internat. J. Numer. Methods Engrg 53, 1801–1829.CrossRefGoogle Scholar
Pang, J. (1990), ‘Newton's method for B-differentiable equations’, Math. Oper. Res. 15, 311–341.CrossRefGoogle Scholar
Pang, J. and Gabriel, S. (1993), ‘NE/SQP: A robust algorithm for the nonlinear complementarity problem’, Math. Progr. 60, 295–337.CrossRefGoogle Scholar
Pang, J. and Qi, L. (1993), ‘Nonsmooth equations: Motivation and algorithms’, SIAM J. Optim. 3, 443–465.CrossRefGoogle Scholar
Pironneau, O. and Achdou, Y. (2009), Partial differential equations for option pricing. In Handbook of Numerical Analysis, Vol XV: Mathematical Modeling and Numerical Methods in Finance (Bensoussan, A.et al, eds), Elsevier/North-Holland, pp. 369–495.CrossRefGoogle Scholar
Popp, A., Gee, M. and Wall, W. (2009), ‘A finite deformation mortar contact formulation using a primal–dual active set strategy’, Internat. J. Numer. Methods Engrg 79, 1354–1391.CrossRefGoogle Scholar
Popp, A., Gitterle, M., Gee, M. and Wall, W. (2010), ‘A dual mortar approach for 3D finite deformation contact with consistent linearization’, Internat. J. Numer. Methods Engrg 83, 1428–1465.CrossRefGoogle Scholar
Pousin, J. and Sassi, T. (2005), ‘A posteriori error estimates and domain decomposition with nonmatching grids’, Adv. Comput. Math. 23, 241–263.CrossRefGoogle Scholar
Prager, W. and Synge, J. (1947), ‘Approximations in elasticity based on concepts of function spaces’, Quart. Appl. Math. 5, 241–269.CrossRefGoogle Scholar
Puso, M. (2004), ‘A 3D mortar method for solid mechanics’, Internat. J. Numer. Methods Engrg 59, 315–336.CrossRefGoogle Scholar
Puso, M. and Laursen, T. (2004 a), ‘A mortar segment-to-segment contact method for large deformation solid mechanics’, Comput. Methods Appl. Mech. Engrg 193, 601–629.CrossRefGoogle Scholar
Puso, M. and Laursen, T. (2004 b), ‘A mortar segment-to-segment frictional contact method for large deformations’, Comput. Methods Appl. Mech. Engrg 193, 4891–4913.CrossRefGoogle Scholar
Puso, M., Laursen, T. and Solberg, J. (2008), ‘A segment-to-segment mortar contact method for quadratic elements and large deformations’, Comput. Methods Appl. Mech. Engrg 197, 555–566.CrossRefGoogle Scholar
Raous, M., Barbarin, S. and Vola, D. (2002), Numerical characterization and computation of dynamic instabilities for frictional contact problems. In Friction and Instabilities (Martinis, J. A. C.et al., ed.), Vol. 457 of CISM Courses Lect., Springer, pp. 233–291.Google Scholar
Raviart, P. and Thomas, J. (1983), Introduction à l'Analyse Numérique des Equations aux Dérivées Partielles, Collection Mathématiques Appliquées pour la Maîtrise, Masson.Google Scholar
Renard, Y. (2006), ‘A uniqueness criterion for the Signorini problem with Coulomb friction’, SIAM J. Math. Anal. 38, 452–467.CrossRefGoogle Scholar
Renard, Y. (2010), ‘The singular dynamic method for constrained second order hyperbolic equations: Application to dynamic contact problems’, J. Comput. Appl. Math. 234, 906–923.CrossRefGoogle Scholar
Repin, S. (2008), A Posteriori Estimates for Partial Differential Equations, Radon Series on Computational and Applied Mathematics, de Gruyter.CrossRefGoogle Scholar
Repin, S., Sauter, S. and Smolianski, A. (2003), ‘A posteriori error estimation for the Dirichlet problem with account of the error in the approximation of boundary conditions’, Computing 70, 205–233.CrossRefGoogle Scholar
Rivière, B. (2008), Discontinuous Galerkin Methods for Solving Elliptic and Parabolic Equations, SIAM.CrossRefGoogle Scholar
Salomon, J., Weiss, A. and Wohlmuth, B. (2008), ‘Energy conserving algorithms for a corotational formulation’, SIAM J. Numer. Anal. 46, 1842–1866.CrossRefGoogle Scholar
Scheidegger, A. (1960), The Physics of Flow through Porous Media, University of Toronto Press.Google Scholar
Schenk, O. and Gärtner, K. (2004), ‘Solving unsymmetric sparse systems of linear equations with PARDISO’, J. Future Generation Computer Systems 20, 475–487.CrossRefGoogle Scholar
Schenk, O. and Gärtner, K. (2006), ‘On fast factorization pivoting methods for symmetric indefinite systems’, Elec. Trans. Numer. Anal. 23, 158–179.Google Scholar
Schöberl, J. (1997), ‘An advancing front 2D/3D-mesh generator based on abstract rules’, Comput. Visual. Sci. 1, 41–52.Google Scholar
Schöberl, J. (1998), ‘Solving the Signorini problem on the basis of domain decomposition techniques’, Computing 60, 323–344.CrossRefGoogle Scholar
Scott, L. and Zhang, S. (1990), ‘Finite element interpolation of nonsmooth functions satisfying boundary conditions’, Math. Comp. 54, 483–493.CrossRefGoogle Scholar
Simeon, B. (2006), ‘On Lagrange multipliers in flexible multibody dynamics’, Comput. Methods Appl. Mech. Engrg 195, 6993–7005.CrossRefGoogle Scholar
Simo, J. (1998), Local behavior in finite element methods. In Numerical Methods for Solids, Part 3 and Numerical Methods for Fluids, Part 1 (Ciarlet, P. and Lions, J., eds), Vol. VI of Handbook of Numerical Analysis, North-Holland, pp. 183–499.CrossRefGoogle Scholar
Simo, J. and Armero, F. (1992), ‘Geometrically nonlinear enhanced strain mixed methods and the method of incompatible modes’, Internat. J. Numer. Methods Engrg 33, 1413–1449.CrossRefGoogle Scholar
Simo, J. and Hughes, T. (1998), Computational Inelasticity, Springer.Google Scholar
Simo, J. and Laursen, T. (1992), ‘Augmented Lagrangian treatment of contact problems involving friction’, Comput. Struct. 42, 97–116.CrossRefGoogle Scholar
Simo, J. and Rifai, M. (1990), ‘A class of assumed strain methods and the method of incompatible modes’, Internat. J. Numer. Methods Engrg 29, 1595–1638.CrossRefGoogle Scholar
Simo, J. and Tarnow, N. (1992), ‘The discrete energy-momentum method: Conserving algorithms for nonlinear elastodynamics’, Z. Angew. Math. Phys. 43, 757–792.CrossRefGoogle Scholar
Simo, J., Armero, F. and Taylor, R. (1993), ‘Improved versions of assumed enhanced trilinear elements for 3D finite deformation problems’, Comput. Methods Appl. Mech. Engrg 110, 359–386.CrossRefGoogle Scholar
Stein, E. and Ohnimus, S. (1997), ‘Equilibrium method for postprocessing and error estimation in the finite element method’, Comput. Assist. Mech. Engrg Sci. 4, 645–666.Google Scholar
Stein, E. and Ohnimus, S. (1999), ‘Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems’, Comput. Methods Appl. Mech. Engrg 176, 363–385.CrossRefGoogle Scholar
Stevenson, R. (2005), ‘An optimal adaptive finite element method’, SIAM J. Numer. Anal. 42, 2188–2217.CrossRefGoogle Scholar
Stevenson, R. (2007), ‘Optimality of a standard adaptive finite element method’, Found. Comput. Math. 7, 245–269.CrossRefGoogle Scholar
Sun, D. and Qi, L. (1999), ‘On NCP-functions’, Comput. Optim. Appl. 13, 201–220.CrossRefGoogle Scholar
Suttmeier, F. (2005), ‘On a direct approach to adaptive FE-discretisations for elliptic variational inequalities’, J. Numer. Math. 13, 73–80.CrossRefGoogle Scholar
Thomée, V. (1997), Galerkin Finite Element Methods for Parabolic Problems, Springer.CrossRefGoogle Scholar
Toselli, A. and Widlund, O. (2005), Domain Decomposition Methods: Algorithms and Theory, Springer.CrossRefGoogle Scholar
Van Genuchten, M. (1980), ‘A closed-form equation for predicting the hydraulic conductivity of unsaturated soils’, Soil Sci. Soc. Am. J. 44, 892–898.CrossRefGoogle Scholar
Veeser, A. (2001), On a posteriori error estimation for constant obstacle problems. In Numerical Methods for Viscosity Solutions and Applications (Falcone, M. and Makridakis, C., eds), Vol. 59 of Advances in Mathematics for Applied Sciences, World Scientific, pp. 221–234.CrossRefGoogle Scholar
Verfürth, R. (1994), ‘A posteriori error estimation and adaptive mesh-refinement techniques’, J. Comput. Appl. Math. 50, 67–83.CrossRefGoogle Scholar
Verfürth, R. (1996), A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Advances in Numerical Mathematics, Wiley–Teubner.Google Scholar
Vohralík, M. (2008), ‘A posteriori error estimation in the conforming finite element method based on its local conservativity and using local minimization’, CR Math. Acad. Sci. Paris 346, 687–690.CrossRefGoogle Scholar
Washizu, K. (1955), On the variational principles of elasticity and plasticity. Report 25–18, Massachusetts Institute of Technology.Google Scholar
Weiss, A. and Wohlmuth, B. (2009), ‘A posteriori error estimator and error control for contact problems’, Math. Comp. 78, 1237–1267.CrossRefGoogle Scholar
Weiss, A. and Wohlmuth, B. (2010), ‘A posteriori error estimator for obstacle problems’, SIAM J. Sci. Comput. 32, 2627–2658.CrossRefGoogle Scholar
Wheeler, M. and Yotov, I. (2005), ‘A posteriori error estimates for the mortar mixed finite element method’, SIAM J. Numer. Anal. 43, 1021–1042.CrossRefGoogle Scholar
Wieners, C. (2007), ‘Nonlinear solution methods for infinitesimal perfect plasticity’, Z. Angew. Math. Mech. 87, 643–660.CrossRefGoogle Scholar
Wieners, C. and Wohlmuth, B. (2011), ‘A primal–dual finite element approximation for a nonlocal model in plasticity’, SIAM J. Sci. Comput. 49, 692–710.Google Scholar
Willner, K. (2003), Kontinuums- und Kontaktmechanik, Springer.CrossRefGoogle Scholar
Wilmott, P., Dewynne, J. and Howison, S. (1997), Option Pricing: Mathematical Models and Computation, Oxford Financial Press.Google Scholar
Wohlmuth, B. (1999 a), ‘Hierarchical a posteriori error estimators for mortar finite element methods with Lagrange multipliers’, SIAM J. Numer. Anal. 36, 1636–1658.CrossRefGoogle Scholar
Wohlmuth, B. (1999 b), ‘A residual based error-estimator for mortar finite element discretizations’, Numer. Math. 84, 143–171.CrossRefGoogle Scholar
Wohlmuth, B. (2000), ‘A mortar finite element method using dual spaces for the Lagrange multiplier’, SIAM J. Numer. Anal. 38, 989–1012.CrossRefGoogle Scholar
Wohlmuth, B. (2001), Discretization Methods and Iterative Solvers Based on Domain Decomposition, Springer.CrossRefGoogle Scholar
Wohlmuth, B. (2005), ‘A V-cycle multigrid approach for mortar finite elements’, SIAM J. Numer. Anal. 42, 2476–2495.CrossRefGoogle Scholar
Wohlmuth, B. (2007), ‘An a posteriori error estimator for two-body contact problems on non-matching meshes’, J. Sci. Comput. 33, 25–45.CrossRefGoogle Scholar
Wohlmuth, B. and Krause, R. (2001), ‘Multigrid methods based on the unconstrained product space for mortar finite element discretizations’, SIAM J. Numer. Anal. 39, 192–213.CrossRefGoogle Scholar
Wohlmuth, B. and Krause, R. (2003), ‘Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems’, SIAM J. Sci. Comput. 25, 324–347.CrossRefGoogle Scholar
Wooding, R. and Morel-Seytoux, H. (1976), ‘Multiphase fluid flow through porous media’, Annu. Rev. Fluid Mech. 8, 233–274.CrossRefGoogle Scholar
Wriggers, P. (2006), Computational Contact Mechanics, second edition, Springer.CrossRefGoogle Scholar
Wriggers, P. and Nackenhorst, U., eds (2007), Computational Methods in Contact Mechanics, Vol. 3 of IUTAM Bookseries, Springer.CrossRefGoogle Scholar
Wriggers, P. and Scherf, O. (1998), ‘Different a posteriori error estimators and indicators for contact problems’, Math. Comput. Modelling 28, 437–447.CrossRefGoogle Scholar
Wright, S. (1997), Primal–Dual Interior Point Methods, SIAM.CrossRefGoogle Scholar
Yang, B. and Laursen, T. (2008 a), ‘A contact searching algorithm including bounding volume trees applied to finite sliding mortar formulations’, Comput. Mech. 41, 189–205.CrossRefGoogle Scholar
Yang, B. and Laursen, T. (2008 b), ‘A large deformation mortar formulation of self contact with finite sliding’, Comput. Methods Appl. Mech. Engrg 197, 756– 772.CrossRefGoogle Scholar
Yang, B., Laursen, T. and Meng, X. (2005), ‘Two dimensional mortar contact methods for large deformation frictional sliding’, Internat. J. Numer. Methods Engrg 62, 1183–1225.CrossRefGoogle Scholar