### Citations

2573 | The Finite Element Method for Elliptic Problems, North-Holland, - Ciarlet - 1978 |

1664 | Optimization and Nonsmooth Analysis - Clarke - 1983 |

1006 | Multi-Grid Methods and Applications - Hackbusch - 1985 |

787 | The Linear Complementarity Problem
- Cottle, Pang, et al.
- 1992
(Show Context)
Citation Context ...cal description of a wide range of phenomena in material science, continuum mechanics, electrodynamics, hydrology and many others. We refer to the monographs of Baiocchi and Capelo [3], Cottle et al. =-=[28]-=-, Duvaut and Lions [32], Glowinski [37] or Kinderlehrer and Stampaccia [50] for an introduction. Even the special case of obstacle problems covers a large and still growing number of applications rang... |

621 |
An Introduction To Variational Inequalities and Their Applications
- Kinderlehrer, Stampacchia
- 1980
(Show Context)
Citation Context ... mechanics, electrodynamics, hydrology and many others. We refer to the monographs of Baiocchi and Capelo [3], Cottle et al. [28], Duvaut and Lions [32], Glowinski [37] or Kinderlehrer and Stampaccia =-=[50]-=- for an introduction. Even the special case of obstacle problems covers a large and still growing number of applications ranging from contact problems in continuum mechanics to option pricing in compu... |

527 |
Multi-level adaptive solutions to boundary-value problems
- Brandt
(Show Context)
Citation Context ... multigrid methods were introduced by Mandel [62, 63] and Kornhuber [51], respectively. A related algorithm by Brandt and Cryer [23] relies on the FAS (full apprioximation storage) approach by Brandt =-=[21]-=-. In order to guarantee that the exact solution of the obstacle problem is a fixed point of the method, they modified the restriction of the residual (but MULTIGRID METHODS FOR OBSTACLE PROBLEMS 27 no... |

436 |
Iterative Methods by Space Decomposition and Subspace Correction
- Xu
- 1992
(Show Context)
Citation Context ...axation (3.10) uν+1j = u ν j + mS∑ l=1 vl, vl = ℓ(λl)− a(wl−1, λl) a(λl, λl) λl, where wl = wl−1 + vl and w0 = u ν j . The following abstract convergence result is a special case of Theorem 4.4 by Xu =-=[85]-=- (see also Xu and Zikatanov [89]). Theorem 3.5. Assume that the splitting (3.7) has the following two properties. There is a constant C0 > 0 such that for all v ∈ Sj there exist vl ∈ Vl satisfying (3.... |

371 |
Numerical Methods for Nonlinear Variational Problems,
- Glowinski
- 1984
(Show Context)
Citation Context ...omena in material science, continuum mechanics, electrodynamics, hydrology and many others. We refer to the monographs of Baiocchi and Capelo [3], Cottle et al. [28], Duvaut and Lions [32], Glowinski =-=[37]-=- or Kinderlehrer and Stampaccia [50] for an introduction. Even the special case of obstacle problems covers a large and still growing number of applications ranging from contact problems in continuum ... |

291 | Introduction to partial differential equations, - Renardy, Rogers - 2004 |

259 |
On the multi-level splitting of finite element spaces.
- Yserentant
- 1986
(Show Context)
Citation Context ...framework as developed by Bramble, Pasciak, Wang, and Xu [18, 19], Bramble and Pasciak [17], Dryja and Widlund [31], Xu [85] and others. The underlying arguments were partly anticipated by Yserentant =-=[90]-=- for the special case of hierarchical splittings (3.21) V0 = I0Sj , Vk = (Ik − Ik−1)Sj , k = 1, . . . , j, where Ik : Sj → Sk denotes nodal interpolation. Using this abstract theory, the most crucial ... |

170 |
Multigrid Methods
- Bramble
- 1993
(Show Context)
Citation Context ...plexity and robustness which make them competitive for certain two–dimensional problems. For further information on successive subspace correction and multigrid, we recommend the monograph of Bramble =-=[16]-=- and the surveys of Xu [85, 87, 88] and Yserentant [91]. The relation of subspace correction and domain decomposition is discussed in some detail by Smith, Bjørstad and Gropp [76]. 4. Subset Decomposi... |

122 |
Guide to multigrid development
- Brandt
- 1982
(Show Context)
Citation Context ...ively for each k = 1, . . . , j by a suitable number of multigrid steps on the preceding level. This procedure is called nested iteration (cf. Hackbusch [39, Chapter 5]) or full multigrid (cf. Brandt =-=[22]-=-). Nested iteration preserves the optimal accuracy ‖u − uj‖ = O(hj). More precisely, starting with ũ0 = u0 and using the stopping criterion (3.17) ‖uk − ũk‖ ≤ σ2 ‖uk − u0k‖ , k = 1, 2, . . . , j, wi... |

121 | Introduction a l’analyse numerique des equations aux derivees partielles, - Raviart, Thomas - 1998 |

117 | The hierarchical basis multigrid method
- BANK, DUPONT, et al.
- 1988
(Show Context)
Citation Context ...rates of associated hierarchical basis multigrid methods deteriorate quadratically and exponentially for two and three space dimensions, respectively (cf. Yserentant [90], Bank, Dupont and Yserentant =-=[4]-=-, Deuflhard, Leinen and Yserentant [30]). On the other hand, hierarchical splittings have some advantages 12 GRÄSER AND KORNHUBER concerning complexity and robustness which make them competitive for ... |

116 |
Convergence estimates for product iterative methods with applications to domain decomposition and multigrid
- Bramble, Pasciak, et al.
(Show Context)
Citation Context ...survey by Xu [88]. Convergence properties of general successive subspace correction methods (cf. Algorithm 3.8) can be analyzed in an abstract framework as developed by Bramble, Pasciak, Wang, and Xu =-=[18, 19]-=-, Bramble and Pasciak [17], Dryja and Widlund [31], Xu [85] and others. The underlying arguments were partly anticipated by Yserentant [90] for the special case of hierarchical splittings (3.21) V0 = ... |

114 |
Convergence estimates for multigrid algorithms without regularity assumptions
- Bramble, Pasciak, et al.
- 1993
(Show Context)
Citation Context ...g steps or W–cycles can be formulated in a similar way. Our heuristic reasoning is confirmed by the celebrated mesh–independent convergence of multigrid methods. Extending the proof of Bramble et al. =-=[18]-=- for symmetric smoothers bk(·, ·) to the actual non–symmetric case, the following convergence result was shown by Neuss [65]. Theorem 3.10. There is a ρ < 1 depending only on the shape regularity of T... |

109 | Multilevel preconditioning
- Dahmen, Kunoth
(Show Context)
Citation Context ... H1–stability of Q0 provides (3.22) with C1 growing linearly in j. Utilizing the equivalence of norms in suitable Besov and Sobolev spaces, (3.22) was first shown by Oswald [67] and Dahmen and Kunoth =-=[29]-=-. For extensions related to adaptively refined grids, we refer to Bornemann and Yserentant [12] and Bramble and Pasciak [17] or Xu [85]. In contrast to the classical multigrid convergence theory of Ha... |

99 | H.: Concepts of an Adaptive Hierarchical Finite Element Code
- Deuflhard, Leinen, et al.
- 1989
(Show Context)
Citation Context ... multigrid methods deteriorate quadratically and exponentially for two and three space dimensions, respectively (cf. Yserentant [90], Bank, Dupont and Yserentant [4], Deuflhard, Leinen and Yserentant =-=[30]-=-). On the other hand, hierarchical splittings have some advantages 12 GRÄSER AND KORNHUBER concerning complexity and robustness which make them competitive for certain two–dimensional problems. For f... |

96 |
Les inequations en mecanique et en physique. Dunod.
- Duvaut, Lions
- 1972
(Show Context)
Citation Context ...de range of phenomena in material science, continuum mechanics, electrodynamics, hydrology and many others. We refer to the monographs of Baiocchi and Capelo [3], Cottle et al. [28], Duvaut and Lions =-=[32]-=-, Glowinski [37] or Kinderlehrer and Stampaccia [50] for an introduction. Even the special case of obstacle problems covers a large and still growing number of applications ranging from contact proble... |

95 |
Domain Decomposition.
- Smith, Bjorstad, et al.
- 1996
(Show Context)
Citation Context ...monograph of Bramble [16] and the surveys of Xu [85, 87, 88] and Yserentant [91]. The relation of subspace correction and domain decomposition is discussed in some detail by Smith, Bjørstad and Gropp =-=[76]-=-. 4. Subset Decomposition Methods We now concentrate on the obstacle problem uj ∈ Kj : J (uj) ≤ J (v) ∀v ∈ Kj as stated in Section 2.1. As minimization over Sj is now replaced by minimization over Kj ... |

95 |
Theory of Multilevel Methods
- Xu
- 1989
(Show Context)
Citation Context ...rojection on Sk. In order to guarantee (3.19) for adaptively refined grids, it is sufficient to choose the subspace Vk = span{Λk \ Λk−1} ⊂ Sk spanned only by the new nodal basis functions (see the Xu =-=[86, 87]-=- and the references cited therein). Straightforward selection Vk = Sk could deteriorate the optimal complexity even up to O(n2j ) in case of strongly local refinement. We come back to the reinterpreta... |

76 | The method of alternating projections and the method of subspace corrections in Hilbert space.
- Xu, Zikatanov
- 2002
(Show Context)
Citation Context ...S∑ l=1 vl, vl = ℓ(λl)− a(wl−1, λl) a(λl, λl) λl, where wl = wl−1 + vl and w0 = u ν j . The following abstract convergence result is a special case of Theorem 4.4 by Xu [85] (see also Xu and Zikatanov =-=[89]-=-). Theorem 3.5. Assume that the splitting (3.7) has the following two properties. There is a constant C0 > 0 such that for all v ∈ Sj there exist vl ∈ Vl satisfying (3.11) v = m∑ l=1 vl, m∑ l=1 ‖vl‖2 ... |

72 | Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids part I: Low order conforming, nonconforming, and mixed FEM.
- Carstensen, Bartels
- 2002
(Show Context)
Citation Context ...teriori estimates of the MULTIGRID METHODS FOR OBSTACLE PROBLEMS 5 discretization error providing appropriate local refinement indicators have been investigated by Veeser [83], Bartels and Carstensen =-=[5]-=-, Kornhuber [53], Braess [13] and others. Nochetto et al. [66] derived a posteriori estimates of the coincidence set Ω• by so–called barrier sets. Convergence proofs for adaptive finite element method... |

70 |
A New Convergence Proof for the Multigrid Method Including the V-cycle
- Braess, Hackbusch
- 1983
(Show Context)
Citation Context ... refined grids, we refer to Bornemann and Yserentant [12] and Bramble and Pasciak [17] or Xu [85]. In contrast to the classical multigrid convergence theory of Hackbusch [38] and Braess and Hackbusch =-=[15]-=- no additional regularity of u is required in order to obtain (3.22) and the resulting mesh–independent convergence (3.16). On the other hand, we get no information how multiple smoothing would improv... |

62 | A basic norm equivalence in the theory of multilevel methods
- Bornemann, Yserentant
- 1978
(Show Context)
Citation Context ... norms in suitable Besov and Sobolev spaces, (3.22) was first shown by Oswald [67] and Dahmen and Kunoth [29]. For extensions related to adaptively refined grids, we refer to Bornemann and Yserentant =-=[12]-=- and Bramble and Pasciak [17] or Xu [85]. In contrast to the classical multigrid convergence theory of Hackbusch [38] and Braess and Hackbusch [15] no additional regularity of u is required in order t... |

59 | Pasciak, New estimates for multilevel algorithms including the
- Bramble, E
- 1991
(Show Context)
Citation Context ... properties of general successive subspace correction methods (cf. Algorithm 3.8) can be analyzed in an abstract framework as developed by Bramble, Pasciak, Wang, and Xu [18, 19], Bramble and Pasciak =-=[17]-=-, Dryja and Widlund [31], Xu [85] and others. The underlying arguments were partly anticipated by Yserentant [90] for the special case of hierarchical splittings (3.21) V0 = I0Sj , Vk = (Ik − Ik−1)Sj ... |

57 |
Formes bilineaires coercitives sur les ensembles convexes,”
- Stampacchia
- 1964
(Show Context)
Citation Context ...ties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set. 1. Introduction Since the pioneering papers of Fichera [35] and Stampaccia =-=[77]-=- almost fifty years ago, variational inequalities have proved extremely useful for the mathematical description of a wide range of phenomena in material science, continuum mechanics, electrodynamics, ... |

55 | Adaptive multilevel-methods in three space dimensions
- BORNEMANN, ERDMANN, et al.
- 1993
(Show Context)
Citation Context ...nds of the interior angles are preserved in course of refinement. Such a stable decomposition of each tetrahedron into eight sub-tetrahedra is more complicated (cf., e.g., Bey [7] or Bornemann et al. =-=[11]-=-). Introducing the step sizes hk, hk = max t∈Tk diam(t), k = 0, . . . , j, 4 GRÄSER AND KORNHUBER we then get (2.7) hj = O(2−j), chk ≤ 12hk−1 ≤ Chk, k = 1, . . . , j, with positive constants c, C ind... |

54 | The cascadic multigrid method for elliptic problems
- Bornemann, Deuflhard
- 1996
(Show Context)
Citation Context ...y are sometimes called optimal. More sophisticated stopping criteria provide optimality even of nested Gauß–Seidel relaxation. This procedure is called cascadic multigrid (cf. Bornemann and Deuflhard =-=[10]-=-), or backslash cycle. The exact finite element solution u0 on the (hopefully) coarse grid T0 can be computed by a direct solver. In order to check the stopping criterion (3.17) a posteriori estimates... |

50 | Monotone multigrid methods for elliptic variational inequalities
- Kornhuber
- 1994
(Show Context)
Citation Context ...on as multigrid V -cycles. We also intend to clarify the relations between different concepts ranging from subset decomposition [78], projected subspace decomposition [1, 2, 63] to monotone multigrid =-=[51]-=- and even active set strategies both with regard to convergence analysis and numerical properties. In particular, we propose a novel truncated nonsmooth Newton multigrid method which can be as well re... |

42 | Adaptive multilevel methods for obstacle problems
- HOPPE, KORNHUBER
- 1994
(Show Context)
Citation Context ...s [43, 82]. As the existing convergence theory typically requires the exact solution of the linear subproblems the combination with inexact (multigrid) solvers is often performed on a heuristic level =-=[48, 49, 61]-=-. In this review we concentrate on extensions of classical multigrid methods to selfadjoint elliptic obstacle problems with box-constraints. Our aim is to bridge the gap between the underlying simple ... |

42 | Rate of convergence of some space decomposition methods foe linear and nonlinear problems
- Tai, Espedal
- 1998
(Show Context)
Citation Context ...tigrid methods even for realistic 3D geometries in biomechanical applications [59]. Projected multilevel relaxation and monotone multigrid has been extended to smooth non-quadratic energy functionals =-=[79, 81]-=- and also to variational inequalities of the form uj ∈ Sj : a(uj , v − uj) + φj(v)− φj(uj) ≥ ℓ(v − uj) ∀v ∈ Sj , with suitable superposition operators φj , see [52, 55]. Applications include frictiona... |

39 |
Error estimates for the approximation of a class of variational inequalities,”
- Falk
- 1974
(Show Context)
Citation Context ...e stability of the continuous free boundary Γ. We refer to Rodrigues [72, Section 6:5] for details. The optimal error estimate ‖u−uj‖ = O(hj) holds for u ∈ H∩H2(Ω), f ∈ L2(Ω), and ϕ ∈ H2(Ω) (cf. Falk =-=[34]-=- or Ciarlet [26, Section 5.1]). First steps towards optimal L2–error estimates have been made by Natterer [64]. Limited regularity of u is reflected by limited order of the discretization error. More ... |

36 | Multilevel additive methods for elliptic finite element problems
- Dryja, Widlund
- 1990
(Show Context)
Citation Context ...uccessive subspace correction methods (cf. Algorithm 3.8) can be analyzed in an abstract framework as developed by Bramble, Pasciak, Wang, and Xu [18, 19], Bramble and Pasciak [17], Dryja and Widlund =-=[31]-=-, Xu [85] and others. The underlying arguments were partly anticipated by Yserentant [90] for the special case of hierarchical splittings (3.21) V0 = I0Sj , Vk = (Ik − Ik−1)Sj , k = 1, . . . , j, wher... |

35 |
Some estimates for a weighted L2 projection
- Bramble, Xu
(Show Context)
Citation Context ...ate the optimal complexity even up to O(n2j ) in case of strongly local refinement. We come back to the reinterpretation of (3.19) in terms of frequencies. It is well–known (cf., e.g., Bramble and Xu =-=[20]-=-) that (3.20) ‖v −Qkv‖L2(Ω) ≤ Chk‖v‖ ∀v ∈ Sj holds with C independent of k and j. As a consequence of (3.20) and an inverse inequality, all functions v ∈ (Qk − Qk−1)2Sj ⊂ (Qk − Qk−1)Sj ⊂ Vk have the p... |

32 |
Error estimates for the finite element solution of variational inequalities
- Brezzi, Hager, et al.
(Show Context)
Citation Context ... of the discretization error. More precisely, even for arbitrarily smooth data the discretization error of piecewise quadratic finite elements only behaves like O(hsj) with s < 1.5 (cf. Brezzi et al. =-=[25]-=-). For many practical problems, in particular in three space dimensions, it is absolutely necessary to use locally refined grids in order to reduce the number of unknowns and therefore the numerical c... |

32 |
Multigrid algorithms for variational inequalities,
- Hoppe
- 1987
(Show Context)
Citation Context ... the coincidence set and a subsequent solution step for the resulting reduced linear problem. This concept has been very popular since the benchmarking work by Hackbusch and Mittelmann [40] and Hoppe =-=[45, 46]-=-. Recent new interest was stimulated by a reinterpretation of the active set approach The authors gratefully acknowledge the support of Xue-Cheng Tai by stimulating discussions and valuable suggestion... |

30 | Multigrid algorithms for the solution of linear complementarity problems arising from free boundary problems,
- Brandt, Cryer
- 1983
(Show Context)
Citation Context ...s to be an open problem. 5.3. Concluding remarks. Standard and truncated multigrid methods were introduced by Mandel [62, 63] and Kornhuber [51], respectively. A related algorithm by Brandt and Cryer =-=[23]-=- relies on the FAS (full apprioximation storage) approach by Brandt [21]. In order to guarantee that the exact solution of the obstacle problem is a fixed point of the method, they modified the restri... |

29 |
Variational and Quasivariational Inequalities,
- Baiocchi, Capelo
- 1984
(Show Context)
Citation Context ...l for the mathematical description of a wide range of phenomena in material science, continuum mechanics, electrodynamics, hydrology and many others. We refer to the monographs of Baiocchi and Capelo =-=[3]-=-, Cottle et al. [28], Duvaut and Lions [32], Glowinski [37] or Kinderlehrer and Stampaccia [50] for an introduction. Even the special case of obstacle problems covers a large and still growing number ... |

29 | Multilevel methods for elliptic problems on domains not resolved by the coarse grid
- Kornhuber, Yserentant
- 1994
(Show Context)
Citation Context ...n of truncated monotone multigrid. Activation/inactivation is performed by a projected Gauß-Seidel step, linear solution is replaced by just one truncated multigrid step (cf. Kornhuber and Yserentant =-=[60]-=-) and global convergence is achieved by damping. Roughly speaking, it turns out that increasing flexibility goes with decreasing theoretical coverage ranging from multigrid convergence rates for multi... |

29 |
A multilevel iterative method for symmetric, positive definite linear complementarity problems
- MANDEL
- 1984
(Show Context)
Citation Context ...iptions of the final implementation as multigrid V -cycles. We also intend to clarify the relations between different concepts ranging from subset decomposition [78], projected subspace decomposition =-=[1, 2, 63]-=- to monotone multigrid [51] and even active set strategies both with regard to convergence analysis and numerical properties. In particular, we propose a novel truncated nonsmooth Newton multigrid met... |

29 |
On function spaces related to finite element approximation theory
- Oswald
- 1990
(Show Context)
Citation Context ...operty (3.20) together with H1–stability of Q0 provides (3.22) with C1 growing linearly in j. Utilizing the equivalence of norms in suitable Besov and Sobolev spaces, (3.22) was first shown by Oswald =-=[67]-=- and Dahmen and Kunoth [29]. For extensions related to adaptively refined grids, we refer to Bornemann and Yserentant [12] and Bramble and Pasciak [17] or Xu [85]. In contrast to the classical multigr... |

29 | Global and uniform convergence of subspace correction methods for some convex optimization problems
- Tai, Xu
- 2002
(Show Context)
Citation Context ...2, 78] for details. Similar to linear subspace decomposition, the abstract convergence result can be also applied to overlapping domain decomposition methods. For further information, we refer to Tai =-=[42, 78, 81]-=- and the references cited therein. 5. Projected subspace decomposition methods 5.1. Projected relaxation methods. Another natural extension of linear subspace correction to obstacle problems is to per... |

29 |
Efficient and reliable A posteriori error estimators for elliptic obstacle problems
- Veeser
- 2002
(Show Context)
Citation Context ...e numerical complexity. A posteriori estimates of the MULTIGRID METHODS FOR OBSTACLE PROBLEMS 5 discretization error providing appropriate local refinement indicators have been investigated by Veeser =-=[83]-=-, Bartels and Carstensen [5], Kornhuber [53], Braess [13] and others. Nochetto et al. [66] derived a posteriori estimates of the coincidence set Ω• by so–called barrier sets. Convergence proofs for ad... |

28 | Convergence rate analysis of an asynchronous space decomposition method for convex minimization,
- Tai, Tseng
- 2001
(Show Context)
Citation Context ...Suttmeier [8]. First results on asymptotic multigrid convergence rates are due to Kornhuber [51]. Major contributions to global bounds for the convergences rates were made by Badea, Tai and coworkers =-=[1, 2, 42, 80]-=-. They also used the same theoretical framework to analyze Jacobi-like versions of Algorithm 5.1, where the update of the intermediate iterates is simply skipped so that the corrections can be compute... |

22 |
and New Convergence Proofs for Multigrid Methods.
- Old
- 1993
(Show Context)
Citation Context ...certain two–dimensional problems. For further information on successive subspace correction and multigrid, we recommend the monograph of Bramble [16] and the surveys of Xu [85, 87, 88] and Yserentant =-=[91]-=-. The relation of subspace correction and domain decomposition is discussed in some detail by Smith, Bjørstad and Gropp [76]. 4. Subset Decomposition Methods We now concentrate on the obstacle problem... |

18 |
Multi{grid convergence theory
- Hackbusch
- 1981
(Show Context)
Citation Context ...orm lower and upper bounds of the algebraic error. 3.3. Concluding remarks. At first sight, our considerations seem to be more complicated than classical approaches to multigrid (cf., e.g., Hackbusch =-=[38]-=-, pp. 17). However, the actual interpretation has the advantage that it suggests direct extensions to obstacle problems later on. We used a very intuitive notion of frequencies. Analytically, the defi... |

18 |
On multigrid methods for variational inequalities
- Hackbusch, Mittelman
- 1983
(Show Context)
Citation Context ...ctual guess for the coincidence set and a subsequent solution step for the resulting reduced linear problem. This concept has been very popular since the benchmarking work by Hackbusch and Mittelmann =-=[40]-=- and Hoppe [45, 46]. Recent new interest was stimulated by a reinterpretation of the active set approach The authors gratefully acknowledge the support of Xue-Cheng Tai by stimulating discussions and ... |

17 |
Finite-Volumen- und Mehrgitterverfahren fur elliptische Randwertprobleme, Ph.D. Thesis, Eberhard-Karls-University Tubingen (in German),
- Bey
- 1997
(Show Context)
Citation Context ...way, lower and upper bounds of the interior angles are preserved in course of refinement. Such a stable decomposition of each tetrahedron into eight sub-tetrahedra is more complicated (cf., e.g., Bey =-=[7]-=- or Bornemann et al. [11]). Introducing the step sizes hk, hk = max t∈Tk diam(t), k = 0, . . . , j, 4 GRÄSER AND KORNHUBER we then get (2.7) hj = O(2−j), chk ≤ 12hk−1 ≤ Chk, k = 1, . . . , j, with po... |

17 |
On multilevel iterative methods for optimization problems
- Gelman, Mandel
- 1990
(Show Context)
Citation Context ... PROBLEMS 19 In order to generalize classical multigrid methods, we now insert the multilevel splitting (3.9) into Algorithm 5.1. The resulting projected multilevel relaxation was suggested by Mandel =-=[36, 62, 63]-=- and later investigated by many authors [1, 2, 42, 51, 81]. The corresponding corrections vl are given explicitly by (5.1) vl = max { rl(λl) a(λl, λl) , max p∈Nj∩int suppλl −wl−1(p) + ϕ(p) λl(p) } λl.... |

16 | Adaptive multigrid methods for Signorini’s problem in linear elasticity
- Kornhuber, Krause
- 2001
(Show Context)
Citation Context ...irections are constant along the Signorini boundary (cf., e.g., Belsky [6]). Spatially varying normal directions can be incorporated by suitable weighting factors as suggested by Kornhuber and Krause =-=[56]-=-. Wohlmuth and Krause [84] extended monotone multigrid to mortar–discretized two-body contact. Their main idea is a hierarchical splitting of the ansatz space into a linear space with vanishing relati... |

16 | Rate of convergence for some constraint decomposition methods for nonlinear variational inequalities
- Tai
(Show Context)
Citation Context ...bspace decomposition and detailed descriptions of the final implementation as multigrid V -cycles. We also intend to clarify the relations between different concepts ranging from subset decomposition =-=[78]-=-, projected subspace decomposition [1, 2, 63] to monotone multigrid [51] and even active set strategies both with regard to convergence analysis and numerical properties. In particular, we propose a n... |

14 |
A posteriori error estimators for obstacle problems—another look, Numer Math 101
- Braess
- 2005
(Show Context)
Citation Context ...TIGRID METHODS FOR OBSTACLE PROBLEMS 5 discretization error providing appropriate local refinement indicators have been investigated by Veeser [83], Bartels and Carstensen [5], Kornhuber [53], Braess =-=[13]-=- and others. Nochetto et al. [66] derived a posteriori estimates of the coincidence set Ω• by so–called barrier sets. Convergence proofs for adaptive finite element methods have been considered by Per... |

14 |
Convergence analysis of a conforming adaptive finite element method for an obstacle problem,
- CARSTENSEN, BRAESS, et al.
- 2007
(Show Context)
Citation Context ...imates of the coincidence set Ω• by so–called barrier sets. Convergence proofs for adaptive finite element methods have been considered by Perez et al. [68], Siebert and Veeser [75] and Braess et al. =-=[14]-=-. 3. Linear Subspace Decomposition Methods 3.1. Spectral properties of elliptic bilinear forms. In this section, we consider the extreme case of an empty coincidence set Ω• = ∅. Obviously, the reduced... |

13 | Nonsmooth Newton-like Methods for Variational Inequalities and Constrained Optimization Problems in Function Spaces
- Ulbrich
- 2002
(Show Context)
Citation Context ...scussions and valuable suggestions. This work has been funded in part by the Deutsche Forschungsgemeinschaft under contract Ko 1806/3-2. 1 2 GRÄSER AND KORNHUBER in terms of nonsmooth Newton methods =-=[43, 82]-=-. As the existing convergence theory typically requires the exact solution of the linear subproblems the combination with inexact (multigrid) solvers is often performed on a heuristic level [48, 49, 6... |

12 | Robust Multigrid Methods for Vector-Valued Allen-Cahn Equations with Logarithmic Free Energy
- Kornhuber, Krause
(Show Context)
Citation Context ...j ∈ Sj : a(uj , v − uj) + φj(v)− φj(uj) ≥ ℓ(v − uj) ∀v ∈ Sj , with suitable superposition operators φj , see [52, 55]. Applications include frictional contact in elasticity [33] or phase field models =-=[57, 58]-=-. 6. From truncated multigrid to inexact active set methods 6.1. A nonsmooth Newton–like method and inexact variants. The truncated monotone multigrid method stated in Algorithm 5.10 has the flavor of... |

11 |
Convergence rate of a Schwarz multilevel method for the constrained minimization of non-quadratic functionals
- Badea
- 2005
(Show Context)
Citation Context ...iptions of the final implementation as multigrid V -cycles. We also intend to clarify the relations between different concepts ranging from subset decomposition [78], projected subspace decomposition =-=[1, 2, 63]-=- to monotone multigrid [51] and even active set strategies both with regard to convergence analysis and numerical properties. In particular, we propose a novel truncated nonsmooth Newton multigrid met... |

11 | On constrained Newton linearization and multigrid for variational inequalities
- Kornhuber
(Show Context)
Citation Context ...rtial differential equations or related variational inequalities can be often traced back to a sequence of obstacle problems playing the same role as linear problems in classical Newton linearization =-=[52, 54, 55, 58]-=-. Finally, apart from their practical relevance, obstacle problems are fascinating mathematical objects of their own value which inherit some, but far from all essential properties from their unconstr... |

11 |
Etude algébrique d’une méthode multigrille pour quelques problèmes de frontière libre
- Mandel
- 1984
(Show Context)
Citation Context ... PROBLEMS 19 In order to generalize classical multigrid methods, we now insert the multilevel splitting (3.9) into Algorithm 5.1. The resulting projected multilevel relaxation was suggested by Mandel =-=[36, 62, 63]-=- and later investigated by many authors [1, 2, 42, 51, 81]. The corresponding corrections vl are given explicitly by (5.1) vl = max { rl(λl) a(λl, λl) , max p∈Nj∩int suppλl −wl−1(p) + ϕ(p) λl(p) } λl.... |

11 |
Fully localized a posteriori error estimators and barrier sets for contact problems
- Nochetto, Siebert, et al.
(Show Context)
Citation Context ...EMS 5 discretization error providing appropriate local refinement indicators have been investigated by Veeser [83], Bartels and Carstensen [5], Kornhuber [53], Braess [13] and others. Nochetto et al. =-=[66]-=- derived a posteriori estimates of the coincidence set Ω• by so–called barrier sets. Convergence proofs for adaptive finite element methods have been considered by Perez et al. [68], Siebert and Veese... |

11 | An introduction to multigrid convergence theory - Xu - 1997 |

10 |
A posteriori error estimates for elliptic variational inequalities
- Kornhuber
- 1996
(Show Context)
Citation Context ...es of the MULTIGRID METHODS FOR OBSTACLE PROBLEMS 5 discretization error providing appropriate local refinement indicators have been investigated by Veeser [83], Bartels and Carstensen [5], Kornhuber =-=[53]-=-, Braess [13] and others. Nochetto et al. [66] derived a posteriori estimates of the coincidence set Ω• by so–called barrier sets. Convergence proofs for adaptive finite element methods have been cons... |

10 | A monotone multigrid solver for two body contact problems in biomechanics
- Kornhuber, Krause, et al.
- 2006
(Show Context)
Citation Context ... the (potential) contact boundary. The resulting algorithm preserves the asymptotic convergence speed of unconstrained multigrid methods even for realistic 3D geometries in biomechanical applications =-=[59]-=-. Projected multilevel relaxation and monotone multigrid has been extended to smooth non-quadratic energy functionals [79, 81] and also to variational inequalities of the form uj ∈ Sj : a(uj , v − uj)... |

10 |
V -cycle convergence with unsymmetric smoothers and application to an anisotropic model problem
- Neuss
- 1998
(Show Context)
Citation Context ...ent convergence of multigrid methods. Extending the proof of Bramble et al. [18] for symmetric smoothers bk(·, ·) to the actual non–symmetric case, the following convergence result was shown by Neuss =-=[65]-=-. Theorem 3.10. There is a ρ < 1 depending only on the shape regularity of T0 and on the ellipticity constant α in (2.4) such that (3.16) ‖uν+1j − uj‖ ≤ ρ‖uνj − uj‖ ∀ν ≥ 0 holds for all u0j ∈ Sj . Mul... |

10 | Solving the Signorini problem on the basis of domain decomposition techniques
- Schöberl
- 1998
(Show Context)
Citation Context ...n methods. For further information, we refer to Badea, Tai and Wang [2] and the references cited therein. A projected space decomposition method of domain-decomposition-type was proposed by Schöberl =-=[74]-=- for Signorini problems in linear elasticity. Exploiting that the number of unknowns in the bulk grows with higher order than the number of unknowns on the boundary he showed mesh-independent converge... |

8 |
The method of subspace corrections
- Xu
- 2001
(Show Context)
Citation Context ... (·, ·) denotes the scalar product in L2. In this sense, the subspaces Vk represent a scale of frequencies. The relation of high frequencies and locality is discussed to some extend in a survey by Xu =-=[88]-=-. Convergence properties of general successive subspace correction methods (cf. Algorithm 3.8) can be analyzed in an abstract framework as developed by Bramble, Pasciak, Wang, and Xu [18, 19], Bramble... |

7 |
The primal-dual active set method as a semi-smooth Newton method
- Hintermüller, Ito, et al.
- 2003
(Show Context)
Citation Context ...scussions and valuable suggestions. This work has been funded in part by the Deutsche Forschungsgemeinschaft under contract Ko 1806/3-2. 1 2 GRÄSER AND KORNHUBER in terms of nonsmooth Newton methods =-=[43, 82]-=-. As the existing convergence theory typically requires the exact solution of the linear subproblems the combination with inexact (multigrid) solvers is often performed on a heuristic level [48, 49, 6... |

7 | B-spline-based monotone multigrid methods,
- Holtz, Kunoth
- 2007
(Show Context)
Citation Context ...number of unknowns nk, k = 2, . . . , 9. The asymptotic convergence rates seem to saturate at about 0.73 (PMLR and SMMG) and 0.41 (TMMG). In comparison with previous computations for generic problems =-=[44, 51]-=-, the complicated coincidence set does not seem to deteriorate the convergence properties of all three methods. 0 20 40 60 80 10010 −15 10−10 10−5 100 105 PMLR SMMG TMMG 0 20 40 60 80 10010 −15 10−10 ... |

7 |
A constrained quadratic minimization with adaptive finite elements. Quaderno n. 13/2005, Dipartimento di Matematica ”F. Enriques”, Universit degli Studi di
- Siebert, Veeser
(Show Context)
Citation Context ...erived a posteriori estimates of the coincidence set Ω• by so–called barrier sets. Convergence proofs for adaptive finite element methods have been considered by Perez et al. [68], Siebert and Veeser =-=[75]-=- and Braess et al. [14]. 3. Linear Subspace Decomposition Methods 3.1. Spectral properties of elliptic bilinear forms. In this section, we consider the extreme case of an empty coincidence set Ω• = ∅.... |

6 | F.T.: A cascadic multigrid algorithm for variational inequalities
- Blum, Braess, et al.
- 2004
(Show Context)
Citation Context ...etimes fails to converge (see Bollrath [9, p. 29] or Kornhuber [51] for a comparison). Preliminary experimental results for cascadic–type iterations have been presented by Blum, Braess, and Suttmeier =-=[8]-=-. First results on asymptotic multigrid convergence rates are due to Kornhuber [51]. Major contributions to global bounds for the convergences rates were made by Badea, Tai and coworkers [1, 2, 42, 80... |

6 |
Rate of convergence for parallel subspace correction methods for nonlinear variational inequalities
- Tai, Heimsund, et al.
- 2002
(Show Context)
Citation Context ...imply skipped. On the other hand, convergence has to be enforced by damping parameters α ≤ nj which might slow down convergence considerably in comparison with the sequential version. We refer to Tai =-=[42, 78]-=- for details. Similar to linear subspace decomposition, the abstract convergence result can be also applied to overlapping domain decomposition methods. For further information, we refer to Tai [42, 7... |

6 |
Two-sided approximations for unilateral variational inequalities by multigrid methods, Optimization 16
- Hoppe
- 1987
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Citation Context ... the coincidence set and a subsequent solution step for the resulting reduced linear problem. This concept has been very popular since the benchmarking work by Hackbusch and Mittelmann [40] and Hoppe =-=[45, 46]-=-. Recent new interest was stimulated by a reinterpretation of the active set approach The authors gratefully acknowledge the support of Xue-Cheng Tai by stimulating discussions and valuable suggestion... |

6 |
Monotone multigrid methods on nonmatching grids for nonlinear multibody contact problems
- Wohlmuth, Krause
- 2003
(Show Context)
Citation Context ...ng the Signorini boundary (cf., e.g., Belsky [6]). Spatially varying normal directions can be incorporated by suitable weighting factors as suggested by Kornhuber and Krause [56]. Wohlmuth and Krause =-=[84]-=- extended monotone multigrid to mortar–discretized two-body contact. Their main idea is a hierarchical splitting of the ansatz space into a linear space with vanishing relative deformation and the (co... |

5 |
A priori error estimates and an inexact primal-dual active set strategy for linear and quadratic finite elements applied to multibody contact problems
- HÜEBER, MAIR, et al.
(Show Context)
Citation Context ...s [43, 82]. As the existing convergence theory typically requires the exact solution of the linear subproblems the combination with inexact (multigrid) solvers is often performed on a heuristic level =-=[48, 49, 61]-=-. In this review we concentrate on extensions of classical multigrid methods to selfadjoint elliptic obstacle problems with box-constraints. Our aim is to bridge the gap between the underlying simple ... |

4 |
Une méthode multigrille pour la solution des problèmes d’obstacle
- Hoppe
- 1990
(Show Context)
Citation Context ...es combined with multigrid are well–known for quite a while. We refer, e.g., to the poineering work of Hackbusch and Mittelmann [40] or the now often called primal dual active set strategies by Hoppe =-=[45, 46, 47]-=-. In contrast to our approach, primal dual active set strategies are essentially iterating on the complementarity condition. For a recent re–interpretation in terms of non–smooth analysis, we refer, e... |

4 |
Multidimensional Coupling in a Human Knee Model
- Sander
- 2008
(Show Context)
Citation Context ...e best choice. Truncated linear multigrid was introduced by Hoppe and Kornhuber [48] and later analyzed by Kornhuber and Yserentant [60]. Numerical experiments for two-body contact problems by Sander =-=[73]-=- confirm the observations of Krause [61] that truncated nonsmooth Newton multigrid 6.2 usually converges faster than truncated monotone multigrid 5.10. 7. Numerical Assessment 7.1. Numerical test prob... |

3 |
A multi–grid method for variational inequalities in contact problems
- Belsky
- 1993
(Show Context)
Citation Context ...for scalar obstacle problems can be also applied directly to Signorini’s problem in linear elasticity, provided that the normal directions are constant along the Signorini boundary (cf., e.g., Belsky =-=[6]-=-). Spatially varying normal directions can be incorporated by suitable weighting factors as suggested by Kornhuber and Krause [56]. Wohlmuth and Krause [84] extended monotone multigrid to mortar–discr... |

3 |
Problemi elastotatici con vincoli unilateral: il problema di Signorini con ambigue condizioni al contorno, Atti
- Fichera
- 1963
(Show Context)
Citation Context ...The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set. 1. Introduction Since the pioneering papers of Fichera =-=[35]-=- and Stampaccia [77] almost fifty years ago, variational inequalities have proved extremely useful for the mathematical description of a wide range of phenomena in material science, continuum mechanic... |

3 |
From inexact active set strategies to nonlinear multigrid methods
- Krause
- 2006
(Show Context)
Citation Context ...s [43, 82]. As the existing convergence theory typically requires the exact solution of the linear subproblems the combination with inexact (multigrid) solvers is often performed on a heuristic level =-=[48, 49, 61]-=-. In this review we concentrate on extensions of classical multigrid methods to selfadjoint elliptic obstacle problems with box-constraints. Our aim is to bridge the gap between the underlying simple ... |

3 |
Optimale L2-Konvergenz finiter Elemente bei Variationsungleichungen
- Natterer
- 1976
(Show Context)
Citation Context ...l error estimate ‖u−uj‖ = O(hj) holds for u ∈ H∩H2(Ω), f ∈ L2(Ω), and ϕ ∈ H2(Ω) (cf. Falk [34] or Ciarlet [26, Section 5.1]). First steps towards optimal L2–error estimates have been made by Natterer =-=[64]-=-. Limited regularity of u is reflected by limited order of the discretization error. More precisely, even for arbitrarily smooth data the discretization error of piecewise quadratic finite elements on... |

2 |
Convergence rate analysis of a multiplicative Schwarz method for variational inequalities
- Tai, Wang
(Show Context)
Citation Context ...Suttmeier [8]. First results on asymptotic multigrid convergence rates are due to Kornhuber [51]. Major contributions to global bounds for the convergences rates were made by Badea, Tai and coworkers =-=[1, 2, 42, 80]-=-. They also used the same theoretical framework to analyze Jacobi-like versions of Algorithm 5.1, where the update of the intermediate iterates is simply skipped so that the corrections can be compute... |

2 | Zwei Mehrgitterverfahren zur numerischen Berechnung von stationären Strömungen durch poröse Medien mit freiem Rand - Bollrath - 1985 |

2 | Solving frictional contact problems with multigrid efficiency
- Fackeldey, Krause
- 2007
(Show Context)
Citation Context ... inequalities of the form uj ∈ Sj : a(uj , v − uj) + φj(v)− φj(uj) ≥ ℓ(v − uj) ∀v ∈ Sj , with suitable superposition operators φj , see [52, 55]. Applications include frictional contact in elasticity =-=[33]-=- or phase field models [57, 58]. 6. From truncated multigrid to inexact active set methods 6.1. A nonsmooth Newton–like method and inexact variants. The truncated monotone multigrid method stated in A... |

2 |
Nonlinear multigrid techniques
- Kornhuber
- 2001
(Show Context)
Citation Context ...rtial differential equations or related variational inequalities can be often traced back to a sequence of obstacle problems playing the same role as linear problems in classical Newton linearization =-=[52, 54, 55, 58]-=-. Finally, apart from their practical relevance, obstacle problems are fascinating mathematical objects of their own value which inherit some, but far from all essential properties from their unconstr... |

1 |
Seuil de régularité pour certaines problèmes unilateraux
- Brézis
- 1971
(Show Context)
Citation Context ...f u jump across the free boundary Γ = Ω•∩Ω◦. Therefore, in contrast to linear elliptic problems, the regularity of u is limited to u ∈ Hs(Ω) with s < 2.5 even for arbitrarily smooth data. See Brézis =-=[24]-=- for a more general result. Regularity or stability of the free boundary Γ is considered in the monograph by Rodrigues [72]. Let us now consider a multilevel finite element discretization of (2.1). A ... |

1 | A numerical adaptive algorithm for the obstacle problem
- Pérez, Cascón, et al.
- 2004
(Show Context)
Citation Context ...s. Nochetto et al. [66] derived a posteriori estimates of the coincidence set Ω• by so–called barrier sets. Convergence proofs for adaptive finite element methods have been considered by Perez et al. =-=[68]-=-, Siebert and Veeser [75] and Braess et al. [14]. 3. Linear Subspace Decomposition Methods 3.1. Spectral properties of elliptic bilinear forms. In this section, we consider the extreme case of an empt... |