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Volumn 27, Issue 1, 2009, Pages 1-44

Review article multigrid methods for obstacle problems

Author keywords

Multigrid methods; Variational inequalities

Indexed keywords


EID: 61349134088     PISSN: 02549409     EISSN: None     Source Type: Journal    
DOI: None     Document Type: Article
Times cited : (71)

References (91)
  • 1
    • 0002219420 scopus 로고
    • Problemi elastotatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno
    • G. Fichera, Problemi elastotatici con vincoli unilaterali: il problema di Signorini con ambigue condizioni al contorno, Atti. Acad. Naz. Lincei Mem. CI. Sci. Fis. Mat. Nat. Sez. la, 7:8 (1963-1964), 91-140.
    • (1963) Atti. Acad. Naz. Lincei Mem. CI. Sci. Fis. Mat. Nat. Sez. la , vol.7 , Issue.8 , pp. 91-140
    • Fichera, G.1
  • 2
    • 0000544120 scopus 로고
    • Formes bilineaires coercitives sur les ensembles convexes
    • G. Stampacchia, Formes bilineaires coercitives sur les ensembles convexes, C.R. Acad. Sci. Paris, 258 (1964), 4413-4416.
    • (1964) C.R. Acad. Sci. Paris , vol.258 , pp. 4413-4416
    • Stampacchia, G.1
  • 5
    • 0003948502 scopus 로고
    • Les inéquations en mécanique et en physique
    • Dunaud, Paris
    • G. Duvaut and J. Lions, Les inéquations en mécanique et en physique, Dunaud, Paris, 1972.
    • (1972)
    • Duvaut, G.1    Lions, J.2
  • 9
    • 0038873931 scopus 로고    scopus 로고
    • Monotone multigrid methods for elliptic variational inequalities II
    • R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities II, Numer. Math., 72 (1996), 481-499.
    • (1996) Numer. Math. , vol.72 , pp. 481-499
    • Kornhuber, R.1
  • 10
    • 61349127657 scopus 로고    scopus 로고
    • Nonlinear multigrid techniques
    • J. Blowey, J. Coleman and A. Craig, editors, Universitext, pages Springer
    • R. Kornhuber, Nonlinear multigrid techniques, J. Blowey, J. Coleman and A. Craig, editors, Theory and Numerics of Differential Equations, Universitext, pages 179-229, Springer, 2001.
    • (2001) Theory and Numerics of Differential Equations , pp. 179-229
    • Kornhuber, R.1
  • 11
    • 0036627648 scopus 로고    scopus 로고
    • On constrained Newton linearization and multigrid for variational inequalities
    • R. Kornhuber, On constrained Newton linearization and multigrid for variational inequalities, Numer. Math., 91 (2002), 699-721.
    • (2002) Numer. Math. , vol.91 , pp. 699-721
    • Kornhuber, R.1
  • 12
    • 33746103660 scopus 로고    scopus 로고
    • Robust multigrid methods for vector-valued Allen-Cahn equations with logarithmic free energy
    • R. Kornhuber and R. Krause, Robust multigrid methods for vector-valued Allen-Cahn equations with logarithmic free energy, Comput. Vis. Sci., 9 (2006), 103-116.
    • (2006) Comput. Vis. Sci. , vol.9 , pp. 103-116
    • Kornhuber, R.1    Krause, R.2
  • 13
    • 0002677045 scopus 로고
    • On multigrid methods for variational inequalities
    • W. Hackbusch and H. Mittelmann, On multigrid methods for variational inequalities, Numer. Math., 42 (1983), 65-76.
    • (1983) Numer. Math. , vol.42 , pp. 65-76
    • Hackbusch, W.1    Mittelmann, H.2
  • 14
    • 0023433751 scopus 로고
    • Multigrid algorithms for variational inequalities
    • R. Hoppe, Multigrid algorithms for variational inequalities, SIAM J. Numer. Anal., 24 (1987), 1046-1065.
    • (1987) SIAM J. Numer. Anal. , vol.24 , pp. 1046-1065
    • Hoppe, R.1
  • 15
    • 0000345936 scopus 로고
    • Two-sided approximations for unilateral variational inequalities by multigrid methods
    • R. Hoppe, Two-sided approximations for unilateral variational inequalities by multigrid methods, Optimization, 18 (1987), 867-881.
    • (1987) Optimization , vol.18 , pp. 867-881
    • Hoppe, R.1
  • 16
    • 0041657515 scopus 로고    scopus 로고
    • The primal-dual active set method as a semi-smooth Newton method
    • M. Hintermüller, K. Ito and K. Kunisch, The primal-dual active set method as a semi-smooth Newton method, SIAM J. Optimiz., 13 (2003), 865-888.
    • (2003) SIAM J. Optimiz. , vol.13 , pp. 865-888
    • Hintermüller, M.1    Ito, K.2    Kunisch, K.3
  • 18
    • 0028420070 scopus 로고
    • Adaptive multilevel methods for obstacle problems
    • R. Hoppe and R. Kornhuber, Adaptive multilevel methods for obstacle problems, SIAM J. Numer. Anal, 31:2 (1994), 301-323.
    • (1994) SIAM J. Numer. Anal , vol.31 , Issue.2 , pp. 301-323
    • Hoppe, R.1    Kornhuber, R.2
  • 19
    • 20444492850 scopus 로고    scopus 로고
    • A priori error estimates and an inexact primal-dual active set strategy for linear and quadratic finite elements applied to multibody contact problems
    • S. Hüeber, M. Mair and B. Wohlmuth, A priori error estimates and an inexact primal-dual active set strategy for linear and quadratic finite elements applied to multibody contact problems., Appl. Num. Math., 54 (2005), 555-576.
    • (2005) Appl. Num. Math. , vol.54 , pp. 555-576
    • Hüeber, S.1    Mair, M.2    Wohlmuth, B.3
  • 21
    • 0242473077 scopus 로고    scopus 로고
    • Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities
    • X.C. Tai, Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities, Numer. Math., 93:4 (2003), 755-786.
    • (2003) Numer. Math. , vol.93 , Issue.4 , pp. 755-786
    • Tai, X.C.1
  • 22
    • 34247183143 scopus 로고    scopus 로고
    • Convergence rate of a Schwarz multilevel method for the constrained minimization of nonquadratic functional
    • L. Badea, Convergence rate of a Schwarz multilevel method for the constrained minimization of nonquadratic functional., SIAM J. Numer. Anal., 44:2 (2006), 449-477.
    • (2006) SIAM J. Numer. Anal. , vol.44 , Issue.2 , pp. 449-477
    • Badea, L.1
  • 23
    • 3142707919 scopus 로고    scopus 로고
    • Convergence rate analysis of a multiplicative Schwarz method for variational inequalities
    • L. Badea, X.C. Tai and J. Wang, Convergence rate analysis of a multiplicative Schwarz method for variational inequalities., SIAM J. Numer. Anal., 41:3 (2003), 1052-1073.
    • (2003) SIAM J. Numer. Anal. , vol.41 , Issue.3 , pp. 1052-1073
    • Badea, L.1    Tai, X.C.2    Wang, J.3
  • 24
    • 0021374981 scopus 로고
    • A multilevel iterative method for symmetric, positive definite linear complementarity problems
    • J. Mandel, A multilevel iterative method for symmetric, positive definite linear complementarity problems, Appl. Math. Opt, 11 (1984), 77-95.
    • (1984) Appl. Math. Opt , vol.11 , pp. 77-95
    • Mandel, J.1
  • 25
    • 21844485573 scopus 로고
    • Monotone multigrid methods for elliptic variational inequalities I
    • R. Kornhuber, Monotone multigrid methods for elliptic variational inequalities I, Numer. Math., 69 (1994), 167-184.
    • (1994) Numer. Math. , vol.69 , pp. 167-184
    • Kornhuber, R.1
  • 26
    • 0012916109 scopus 로고
    • Multilevel methods for elliptic problems on domains not resolved by the coarse grid
    • R. Kornhuber and H. Yserentant, Multilevel methods for elliptic problems on domains not resolved by the coarse grid, Contemporary Mathematics, 180 (1994), 49-60.
    • (1994) Contemporary Mathematics , vol.180 , pp. 49-60
    • Kornhuber, R.1    Yserentant, H.2
  • 27
    • 39849107405 scopus 로고
    • Seuil de régularité pour certaines problèmes unilateraux
    • H. Brézis, Seuil de régularité pour certaines problèmes unilateraux, C.R. Ac. Sci. Paris, 273 (1971), 35-37.
    • (1971) C.R. Ac. Sci. Paris , vol.273 , pp. 35-37
    • Brézis, H.1
  • 28
    • 2942560020 scopus 로고    scopus 로고
    • Finite-Volumen- und Mehrgitterverfahren für elliptische Randwertprobleme
    • Stuttgart
    • J. Bey, Finite-Volumen- und Mehrgitterverfahren für elliptische Randwertprobleme, Teubner, Stuttgart, 1998.
    • (1998) Teubner
    • Bey, J.1
  • 30
    • 0000003984 scopus 로고
    • Error estimates for the approximation of a class of variational inequalities
    • R. Falk, Error estimates for the approximation of a class of variational inequalities, Math. Comput., 28 (1974), 963-971.
    • (1974) Math. Comput. , vol.28 , pp. 963-971
    • Falk, R.1
  • 32
    • 0001907381 scopus 로고
    • 2-Konvergenz finiter Elemente bei Variationsungleichungen
    • 2-Konvergenz finiter Elemente bei Variationsungleichungen, Bonn. Math. Schr., 89 (1976), 1-12.
    • (1976) Bonn. Math. Schr. , vol.89 , pp. 1-12
    • Natterer, F.1
  • 33
    • 0000947683 scopus 로고
    • Error estimates for the finite element solution of variational inequalities I
    • F. Brezzi, W. Hager and P. Raviart, Error estimates for the finite element solution of variational inequalities I, Numer. Math., 28 (1977), 431-443.
    • (1977) Numer. Math. , vol.28 , pp. 431-443
    • Brezzi, F.1    Hager, W.2    Raviart, P.3
  • 34
    • 0035733517 scopus 로고    scopus 로고
    • Efficient and reliable a posteriori error estimators for elliptic obstacle problems
    • A. Veeser, Efficient and reliable a posteriori error estimators for elliptic obstacle problems, SIAM J. Numer. Anal., 39 (2001), 146-167.
    • (2001) SIAM J. Numer. Anal. , vol.39 , pp. 146-167
    • Veeser, A.1
  • 35
    • 11144295563 scopus 로고    scopus 로고
    • Averaging techniques yield reliable a posteriori finite element error control for obstacle problems
    • S. Bartels and C. Carstensen, Averaging techniques yield reliable a posteriori finite element error control for obstacle problems, Numer. Math., 99:2 (2004), 225-249.
    • (2004) Numer. Math. , vol.99 , Issue.2 , pp. 225-249
    • Bartels, S.1    Carstensen, C.2
  • 36
    • 0030128859 scopus 로고    scopus 로고
    • A posteriori error estimates for elliptic variational inequalities
    • R. Kornhuber, A posteriori error estimates for elliptic variational inequalities, Comput. Math. Appl., 31 (1996), 49-60.
    • (1996) Comput. Math. Appl. , vol.31 , pp. 49-60
    • Kornhuber, R.1
  • 37
    • 29144485659 scopus 로고    scopus 로고
    • A posteriori error estimators for obstacle problems-another look
    • D. Braess, A posteriori error estimators for obstacle problems-another look., Numer. Math., 101 (2005), 415-421.
    • (2005) Numer. Math. , vol.101 , pp. 101
    • Braess, D.1
  • 38
    • 27744463727 scopus 로고    scopus 로고
    • Fully localized a posteriori error estimators and barrier sets for contact problems
    • R. Nochetto, K. Siebert and A. Veeser, Fully localized a posteriori error estimators and barrier sets for contact problems, SIAM J. Numer. Anal., 42 (2005), 2118-2135.
    • (2005) SIAM J. Numer. Anal. , vol.42 , pp. 2118-2135
    • Nochetto, R.1    Siebert, K.2    Veeser, A.3
  • 39
    • 35048855568 scopus 로고    scopus 로고
    • A numerical adaptive algorithm for the obstacle problem
    • M. Bubak, editor, Lecture Notes in Computer Science, Springer
    • F. Pérez, J. Cascon and L. Ferragut, A numerical adaptive algorithm for the obstacle problem., M. Bubak, editor, Proceedings of Computational science-IOCS, Part II, volume 3037 of Lecture Notes in Computer Science, pages 130-137, Springer, 2004.
    • (2004) Proceedings of Computational science-IOCS, Part II , vol.3037 , pp. 130-137
    • Pérez, F.1    Cascon, J.2    Ferragut, L.3
  • 40
    • 39449115451 scopus 로고    scopus 로고
    • A unilaterally constrained quadratic minimization with adaptive finite elements
    • K. Siebert and A. Veeser, A unilaterally constrained quadratic minimization with adaptive finite elements., SIAM J. Optimiz., 18 (2007), 260-289.
    • (2007) SIAM J. Optimiz. , vol.18 , pp. 260-289
    • Siebert, K.1    Veeser, A.2
  • 41
    • 34548069658 scopus 로고    scopus 로고
    • Convergence analysis of a conforming adaptive finite element method for an obstacle problem
    • D. Braess, C. Carstensen and R. Hoppe, Convergence analysis of a conforming adaptive finite element method for an obstacle problem., Numer. Math., 107 (2007), 455-471.
    • (2007) Numer. Math. , vol.107 , pp. 455-471
    • Braess, D.1    Carstensen, C.2    Hoppe, R.3
  • 42
    • 61349141719 scopus 로고
    • Introduction a l'analyse numerique des equations aux derivees partielles
    • Paris
    • P. Raviart and J. Thomas, Introduction a l'analyse numerique des equations aux derivees partielles, Masson, Paris, 1992.
    • (1992) Masson
    • Raviart, P.1    Thomas, J.2
  • 46
    • 0000570696 scopus 로고
    • Iterative methods by space decomposition and subspace correction
    • J. Xu, Iterative methods by space decomposition and subspace correction, SIAM Rev., 34 (1992), 581-613.
    • (1992) SIAM Rev. , vol.34 , pp. 581-613
    • Xu, J.1
  • 47
    • 0036016141 scopus 로고    scopus 로고
    • The method of alternating projections and the method of subspace corrections in Hilbert space
    • J. Xu and L. Zikatanov, The method of alternating projections and the method of subspace corrections in Hilbert space., J. Am. Math. Soc., 15:3 (2002), 573-597.
    • (2002) J. Am. Math. Soc. , vol.15 , Issue.3 , pp. 573-597
    • Xu, J.1    Zikatanov, L.2
  • 48
    • 34250126304 scopus 로고
    • On the multi-level splitting of finite element spaces
    • H. Yserentant, On the multi-level splitting of finite element spaces, Numer. Math., 49 (1986), 379-412.
    • (1986) Numer. Math. , vol.49 , pp. 379-412
    • Yserentant, H.1
  • 49
    • 84966255975 scopus 로고
    • Convergence estimates for multigrid algorithms without regularity assumptions
    • J. Bramble, J. Pasciak, J. Wang and J. Xu, Convergence estimates for multigrid algorithms without regularity assumptions, Math. Comput., 57 (1991), 23-45.
    • (1991) Math. Comput. , vol.57 , pp. 23-45
    • Bramble, J.1    Pasciak, J.2    Wang, J.3    Xu, J.4
  • 50
    • 0000696868 scopus 로고    scopus 로고
    • V-cycle convergence with unsymmetric smoothers and application to an anisotropic model problem
    • N. Neuss, V-cycle convergence with unsymmetric smoothers and application to an anisotropic model problem, SIAM J. Numer. Anal, 35 (1998), 1201-1212.
    • (1998) SIAM J. Numer. Anal , vol.35 , pp. 1201-1212
    • Neuss, N.1
  • 52
    • 0030369254 scopus 로고    scopus 로고
    • The cascadic multigrid method for elliptic problems
    • F. Bornemann and P. Deuflhard, The cascadic multigrid method for elliptic problems, Numer. Math., 75 (1996), 135-152.
    • (1996) Numer. Math. , vol.75 , pp. 135-152
    • Bornemann, F.1    Deuflhard, P.2
  • 54
    • 2942528114 scopus 로고    scopus 로고
    • An introduction to multigrid convergence theory
    • R.H. Chan et al., editor, Springer
    • J. Xu, An introduction to multigrid convergence theory, R.H. Chan et al., editor, Iterative Methods in Scientific Computing, pages 169-241, Springer, 1997.
    • (1997) Iterative Methods in Scientific Computing , pp. 169-241
    • Xu, J.1
  • 55
    • 0001462412 scopus 로고    scopus 로고
    • An introduction to multilevel methods
    • M. Ainsworth et al., editor, Oxford, Clarendon Press
    • J. Xu, An introduction to multilevel methods, M. Ainsworth et al., editor, Wavelets, Multilevel Methods and Elliptic PDEs, pages 213-302, Oxford, 1997, Clarendon Press.
    • (1997) Wavelets, Multilevel Methods and Elliptic PDEs , pp. 213-302
    • Xu, J.1
  • 56
    • 84966231734 scopus 로고
    • Some estimates for a weighted L2 projection
    • J. Bramble and J. Xu, Some estimates for a weighted L2 projection, Math. Comput., 56 (1991), 463-476.
    • (1991) Math. Comput. , vol.56 , pp. 463-476
    • Bramble, J.1    Xu, J.2
  • 57
    • 0035281551 scopus 로고    scopus 로고
    • The method of subspace corrections
    • J. Xu, The method of subspace corrections, J. Comput. Appl. Math., 128 (2001), 335-362.
    • (2001) J. Comput. Appl. Math. , vol.128 , pp. 335-362
    • Xu, J.1
  • 58
    • 84966225192 scopus 로고
    • Convergence estimates for product iterative methods with application to domain decomposition
    • J. Bramble, J. Pasciak, J. Wang and J. Xu, Convergence estimates for product iterative methods with application to domain decomposition, Math. Comput., 57 (1991), 1-21.
    • (1991) Math. Comput. , vol.57 , pp. 1-21
    • Bramble, J.1    Pasciak, J.2    Wang, J.3    Xu, J.4
  • 59
    • 84968468806 scopus 로고
    • New estimates for multilevel algorithms including the V-cycle
    • J. Bramble and J. Pasciak, New estimates for multilevel algorithms including the V-cycle, Math. Comput, 60 (1993), 447-471.
    • (1993) Math. Comput , vol.60 , pp. 447-471
    • Bramble, J.1    Pasciak, J.2
  • 60
    • 0002990325 scopus 로고
    • Multilevel additive methods for elliptic finite element problems
    • W. Hackbusch, editor, Braunschweig, Vieweg
    • M. Dryja and O. Widlund, Multilevel additive methods for elliptic finite element problems, W. Hackbusch, editor, Parallel Algorithms for Partial Differential Equations, Braunschweig, 1991, Vieweg.
    • (1991) Parallel Algorithms for Partial Differential Equations
    • Dryja, M.1    Widlund, O.2
  • 61
    • 0001988435 scopus 로고
    • On function spaces related to finite element approximation theory
    • P. Oswald, On function spaces related to finite element approximation theory, Z. Anal. Anwend., 9 (1990), 43-64.
    • (1990) Z. Anal. Anwend. , vol.9 , pp. 43-64
    • Oswald, P.1
  • 62
    • 33845592412 scopus 로고
    • Multilevel preconditioning
    • W. Dahmen and A. Kunoth, Multilevel preconditioning, Numer. Math., 63 (1992), 315-344.
    • (1992) Numer. Math. , vol.63 , pp. 315-344
    • Dahmen, W.1    Kunoth, A.2
  • 63
    • 21144473187 scopus 로고
    • A basic norm equivalence in the theory of multilevel methods
    • F. Bornemann and H. Yserentant, A basic norm equivalence in the theory of multilevel methods, Numer. Math., 64 (1993), 455-476.
    • (1993) Numer. Math. , vol.64 , pp. 455-476
    • Bornemann, F.1    Yserentant, H.2
  • 64
    • 0000671861 scopus 로고
    • A new convergence proof for the multigrid method including the V-cycle
    • D. Braess and W. Hackbusch, A new convergence proof for the multigrid method including the V-cycle, SIAM J. Numer. Anal, 20 (1983), 967-975.
    • (1983) SIAM J. Numer. Anal , vol.20 , pp. 967-975
    • Braess, D.1    Hackbusch, W.2
  • 65
    • 0001377512 scopus 로고
    • The hierarchical basis multigrid method
    • R. Bank, T. Dupont and H. Yserentant, The hierarchical basis multigrid method, Numer. Math., 52 (1988), 387-404.
    • (1988) Numer. Math. , vol.52 , pp. 387-404
    • Bank, R.1    Dupont, T.2    Yserentant, H.3
  • 68
    • 84971922635 scopus 로고
    • Old and new convergence proofs for multigrid methods
    • H. Yserentant, Old and new convergence proofs for multigrid methods, Acta Numerica, (1993), 285-326.
    • (1993) Acta Numerica , pp. 285-326
    • Yserentant, H.1
  • 70
    • 79957958876 scopus 로고    scopus 로고
    • Rate of convergence for parallel subspace correction methods for nonlinear variational inequalities
    • N.D.et al., editor, CIMNE
    • B. Heimsund, X.C. Tai and J. Xu, Rate of convergence for parallel subspace correction methods for nonlinear variational inequalities, N.D.et al., editor, Domain Decomposition Methods in Science and Engineering, pages 127-138, CIMNE, 2002.
    • (2002) Domain Decomposition Methods in Science and Engineering , pp. 127-138
    • Heimsund, B.1    Tai, X.C.2    Xu, J.3
  • 71
    • 0036003411 scopus 로고    scopus 로고
    • Global and uniform convergence of subspace correction methods for some convex optimization problems
    • X.C. Tai and J. Xu, Global and uniform convergence of subspace correction methods for some convex optimization problems, Math. Comput., 71 (2001), 105-124.
    • (2001) Math. Comput. , vol.71 , pp. 105-124
    • Tai, X.C.1    Xu, J.2
  • 72
    • 0025403695 scopus 로고
    • On multilevel iterative method for optimization problems
    • E. Gelman and J. Mandel, On multilevel iterative method for optimization problems, Math. Program., 48 (1990), 1-17.
    • (1990) Math. Program. , vol.48 , pp. 1-17
    • Gelman, E.1    Mandel, J.2
  • 73
    • 0040229571 scopus 로고
    • Etude algebrique d'une methode multigrille pour quelques problèmes de frontière libre
    • J. Mandel, Etude algebrique d'une methode multigrille pour quelques problèmes de frontière libre, C.R. Acad. Sci., 298 (1984), 469-472, Ser. I.
    • (1984) C.R. Acad. Sci. Ser. I , vol.298 , pp. 469-472
    • Mandel, J.1
  • 74
    • 0000815547 scopus 로고
    • Multigrid algorithms for the solution of linear complementary problems arising from free boundary problems
    • A. Brandt and C. Cryer, Multigrid algorithms for the solution of linear complementary problems arising from free boundary problems, SIAM J. Sci. Stat. Comput, 4 (1983), 655-684.
    • (1983) SIAM J. Sci. Stat. Comput , vol.4 , pp. 655-684
    • Brandt, A.1    Cryer, C.2
  • 75
    • 84966233568 scopus 로고
    • Multi-level adaptive solutions to boundary-value problems
    • A. Brandt, Multi-level adaptive solutions to boundary-value problems, Math. Comput., 31 (1977), 333-390.
    • (1977) Math. Comput. , vol.31 , pp. 333-390
    • Brandt, A.1
  • 76
    • 61349163453 scopus 로고
    • Zwei Mehrgitterverfahren zur numerischen Berechnung von stationaren Stromungen durch pörose Medien mit freiem Rand
    • PhD Thesis, Universität Bochum
    • C. Bollrath, Zwei Mehrgitterverfahren zur numerischen Berechnung von stationaren Stromungen durch pörose Medien mit freiem Rand, PhD Thesis, Universität Bochum, 1985.
    • (1985)
    • Bollrath, C.1
  • 77
    • 0036018013 scopus 로고    scopus 로고
    • Convergence rate analysis of an asynchronous space decomposition method for convex minimization
    • X.C. Tai and P. Tseng, Convergence rate analysis of an asynchronous space decomposition method for convex minimization, Math. Comput, 71 (2001), 1105-1135.
    • (2001) Math. Comput , vol.71 , pp. 1105-1135
    • Tai, X.C.1    Tseng, P.2
  • 78
    • 34548280428 scopus 로고    scopus 로고
    • A cascadic multigrid algorithm for variational inequalities
    • H. Blum, D. Braess and F. Suttmeier, A cascadic multigrid algorithm for variational inequalities., Comput. Vis. Sci., 7 (2004), 153-157.
    • (2004) Comput. Vis. Sci. , vol.7 , pp. 153-157
    • Blum, H.1    Braess, D.2    Suttmeier, F.3
  • 79
    • 0031643878 scopus 로고    scopus 로고
    • Solving the Signorini problem on the basis of domain decomposition techniques
    • J. Schoberl, Solving the Signorini problem on the basis of domain decomposition techniques, Computing, 60:4 (1998), 323-344.
    • (1998) Computing , vol.60 , Issue.4 , pp. 323-344
    • Schoberl, J.1
  • 80
    • 34250076098 scopus 로고
    • A multi-grid method for variational inequalities in contact problems
    • V. Belsky, A multi-grid method for variational inequalities in contact problems., Computing, 51 (1993), 293-311.
    • (1993) Computing , vol.51 , pp. 293-311
    • Belsky, V.1
  • 81
    • 0242507151 scopus 로고    scopus 로고
    • Adaptive multigrid methods for Signorini's problem in linear elasticity
    • R. Kornhuber and R. Krause, Adaptive multigrid methods for Signorini's problem in linear elasticity, Comp. Visual. Sci., 4 (2001), 9-20.
    • (2001) Comp. Visual. Sci. , vol.4 , pp. 9-20
    • Kornhuber, R.1    Krause, R.2
  • 82
    • 1842564097 scopus 로고    scopus 로고
    • Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems
    • B. Wohlmuth and R. Krause, Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems., SIAM J. Sci. Comput, 25 (2003), 324-347.
    • (2003) SIAM J. Sci. Comput , vol.25 , pp. 324-347
    • Wohlmuth, B.1    Krause, R.2
  • 83
    • 37249075094 scopus 로고    scopus 로고
    • A monotone multigrid solver for two body contact problems in biomechanics.
    • R. Kornhuber, R. Krause, O. Sander, P. Deuflhard and S. Ertel, A monotone multigrid solver for two body contact problems in biomechanics., Comp. Visual. Sci., 11 (2008), 3-15.
    • (2008) Comp. Visual. Sci. , vol.11 , pp. 3-15
    • Kornhuber, R.1    Krause, R.2    Sander, O.3    Deuflhard, P.4    Ertel, S.5
  • 84
    • 0000986730 scopus 로고    scopus 로고
    • Rate of convergence of some space decomposition method for linear and non-linear elliptic problems
    • X.C. Tai and M.S. Espedal, Rate of convergence of some space decomposition method for linear and non-linear elliptic problems, SIAM J. Numer. Anal., 35 (1998), 1558-1570.
    • (1998) SIAM J. Numer. Anal. , vol.35 , pp. 1558-1570
    • Tai, X.C.1    Espedal, M.S.2
  • 85
    • 61349174436 scopus 로고    scopus 로고
    • Solving frictional contact problems with multigrid efficiency.
    • O. Widlund and D. Keyes, editors, Springer
    • K. Fackeldey and R. Krause, Solving frictional contact problems with multigrid efficiency., O. Widlund and D. Keyes, editors, Domain Decomposition Methods in Science and Engineering, volume 55 of LNCSE, pages 549-556, Springer, 2006.
    • (2006) Domain Decomposition Methods in Science and Engineering , vol.55 , pp. 549-556
    • Kornhuber, R.1    Krause, R.2
  • 86
    • 19844374008 scopus 로고    scopus 로고
    • On multigrid methods for vector-valued Allen-Cahn equations
    • I. Herrera et al., editor, Mexico City, Mexico, UNAM
    • R. Kornhuber and R. Krause, On multigrid methods for vector-valued Allen-Cahn equations, I. Herrera et al., editor, Domain Decomposition Methods in Science and Engineering, pages 307-314, Mexico City, Mexico, 2003, UNAM.
    • (2003) Domain Decomposition Methods in Science and Engineering , pp. 307-314
    • Kornhuber, R.1    Krause, R.2
  • 88
    • 0039251771 scopus 로고
    • Une méthode multigrille pour la solution des problèmes d'obstacle
    • 2 AN, 24 (1990), 711-736.
    • (1990) 2 AN , vol.24 , pp. 711-736
    • Hoppe, R.1
  • 89
    • 61349152829 scopus 로고    scopus 로고
    • Multidimensional coupling in a human knee model
    • PhD Thesis, FU Berlin
    • O. Sander, Multidimensional Coupling in a Human Knee Model, PhD Thesis, FU Berlin, 2008.
    • (2008)
    • Sander, O.1
  • 90
    • 46949091375 scopus 로고    scopus 로고
    • B-spline-based monotone multigrid methods
    • M. Holtz and A. Kunoth, B-spline-based monotone multigrid methods, SIAM J. Numer. Anal., 45 (2007), 1175-1199.
    • (2007) SIAM J. Numer. Anal. , vol.45 , pp. 1175-1199
    • Holtz, M.1    Kunoth, A.2
  • 91
    • 0004246507 scopus 로고
    • Lecture Notes in Mathematics, Berlin, Springer
    • W. Hackbusch and U. Trottenberg, editors, Multigrid Methods, volume 960 of Lecture Notes in Mathematics, Berlin, 1982, Springer.
    • (1982) Multigrid Methods , vol.960
    • Hackbusch, W.1    Trottenberg, U.2


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