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The Auxiliary Space Method And Optimal Multigrid Preconditioning Techniques For Unstructured Grids
 Computing
, 1996
"... . An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxi ..."
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Cited by 55 (6 self)
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. An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a further nested multigrid method can be naturally applied. This new technique make it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris ...
Monotone Multigrid Methods for Elliptic Variational Inequalities I
 I. Numer. Math
, 1993
"... . We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation ..."
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Cited by 50 (13 self)
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. We derive fast solvers for discrete elliptic variational inequalities of the first kind (obstacle problems) as resulting from the approximation of related continuous problems by piecewise linear finite elements. Using basic ideas of successive subspace correction, we modify wellknown relaxation methods by extending the set of search directions. Extended underrelaxations are called monotone multigrid methods, if they are quasioptimal in a certain sense. By construction, all monotone multigrid methods are globally convergent. We take a closer look at two natural variants, the standard monotone multigrid method and a truncated version. For the considered model problems, the asymptotic convergence rates resulting from the standard approach suffer from insufficient coarsegrid transport, while the truncated monotone multigrid method provides the same efficiency as in the unconstrained case. Key words: obstacle problems, adaptive finite element methods, multigrid methods AMS (MOS) subje...
Boundary Treatments For Multilevel Methods On Unstructured Meshes
, 1996
"... . In applying multilevel iterative methods on unstructured meshes, the grid hierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid. In this case, the standard coarsetofine grid transfer operators with linear interpolants are not accurate enoug ..."
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Cited by 15 (8 self)
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. In applying multilevel iterative methods on unstructured meshes, the grid hierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid. In this case, the standard coarsetofine grid transfer operators with linear interpolants are not accurate enough at Neumann boundaries so special care is needed to correctly handle di#erent types of boundary conditions. We propose two e#ective ways to adapt the standard coarsetofine interpolations to correctly implement boundary conditions for twodimensional polygonal domains, and we provide some numerical examples of multilevel Schwarz methods (and multigrid methods) which show that these methods are as e#cient as in the structured case. In addition, we prove that the proposed interpolants possess the local optimal L 2 approximation and H 1 stability, which are essential in the convergence analysis of the multilevel Schwarz methods. Using these results, we give a condition number bound for ...
Multigrid methods for obstacle problems
"... Abstract. In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which ..."
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Cited by 14 (2 self)
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Abstract. In this review, we intend to clarify the underlying ideas and the relations between various multigrid methods ranging from subset decomposition, to projected subspace decomposition and truncated multigrid. In addition, we present a novel globally convergent inexact active set method which is closely related to truncated multigrid. The numerical properties of algorithms are carefully assessed by means of a degenerate problem and a problem with a complicated coincidence set. 1.
Parallel Adaptive Subspace Correction Schemes with Applications to Elasticity
 Comput. Methods Appl. Mech. Engrg
, 1999
"... : In this paper, we give a survey on the three main aspects of the efficient treatment of PDEs, i.e. adaptive discretization, multilevel solution and parallelization. We emphasize the abstract approach of subspace correction schemes and summarize its convergence theory. Then, we give the main featur ..."
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Cited by 11 (4 self)
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: In this paper, we give a survey on the three main aspects of the efficient treatment of PDEs, i.e. adaptive discretization, multilevel solution and parallelization. We emphasize the abstract approach of subspace correction schemes and summarize its convergence theory. Then, we give the main features of each of the three distinct topics and treat the historical background and modern developments. Furthermore, we demonstrate how all three ingredients can be put together to give an adaptive and parallel multilevel approach for the solution of elliptic PDEs and especially of linear elasticity problems. We report on numerical experiments for the adaptive parallel multilevel solution of some test problems, namely the Poisson equation and Lam'e's equation. Here, we emphasize the parallel efficiency of the adaptive code even for simple test problems with little work to distribute, which is achieved through hash storage techniques and spacefilling curves. Keywords: subspace correction, iter...
On Constrained Newton Linearization And Multigrid For Variational Inequalities
"... . We consider the fast solution of a class of large, piecewise smooth minimization problems. For lack of smoothness, usual Newton multigrid methods cannot be applied. We propose a new approach based on a combination of convex minization with constrained Newton linearization. No regularization is inv ..."
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Cited by 11 (5 self)
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. We consider the fast solution of a class of large, piecewise smooth minimization problems. For lack of smoothness, usual Newton multigrid methods cannot be applied. We propose a new approach based on a combination of convex minization with constrained Newton linearization. No regularization is involved. We show global convergence of the resulting monotone multigrid methods and give polylogarithmic upper bounds for the asymptotic convergence rates. Eciency is illustrated by numerical experiments. 1. Introduction Let be a bounded, polyhedral domain in the Euclidean space R d . We consider the minimization problem u 2 H : J (u) + (u) J (v) + (v) 8v 2 H (1.1) on a closed subspace H H 1( 2 For simplicity, we concentrate on H = H 1 0 and d = 2. The quadratic functional J , J (v) = 1 2 a(v; v) `(v); (1.2) is induced by a continuous, symmetric and H{elliptic bilinear form a(; ) and by a linear functional ` 2 H 0 . H is equipped with the energy norm k k = a(; ) 1=2 ...
A monotone multigrid solver for two body contact problems in biomechanics
 Comput. Vis. Sci
"... Abstract. The purpose of the paper is to apply monotone multigrid methods to static and dynamic biomechanical contact problems. In space, a finite element method involving a mortar discretization of the contact conditions is used. In time, a new contact–stabilized Newmark scheme is presented. Numer ..."
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Cited by 10 (7 self)
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Abstract. The purpose of the paper is to apply monotone multigrid methods to static and dynamic biomechanical contact problems. In space, a finite element method involving a mortar discretization of the contact conditions is used. In time, a new contact–stabilized Newmark scheme is presented. Numerical experiments for a two body Hertzian contact problem and a biomechanical application are reported. 1.
Twoscale Composite Finite Element method for the Dirichlet problem on complicated domains. Numerische Mathematik
"... In this paper, we will define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for ..."
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Cited by 8 (4 self)
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In this paper, we will define a new class of finite elements for the discretization of problems with Dirichlet boundary conditions. In contrast to standard finite elements, the minimal dimension of the approximation space is independent of the domain geometry and this is especially advantageous for problems on domains with complicated microstructures. For the proposed finite element method we prove the optimalorder approximation (up to logarithmic terms) and convergence estimates valid also in the cases when the exact solution has a reduced regularity due to reentering corners of the domain boundary.
Hierarchical Basis for the ConvectionDiffusion Equation on Unstructured Meshes
 in Ninth International Symposium on Domain Decomposition Methods for Partial Differential Equations
, 1997
"... Introduction The Hierarchical Basis Multigrid Method was originally developed for sequences of refined meshes. Hierarchical basis functions can be constructed in a straightforward fashion on such sequences of nested meshes. The HBMG iteration itself is just a block symmetric GauSeidel iteration ap ..."
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Cited by 6 (2 self)
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Introduction The Hierarchical Basis Multigrid Method was originally developed for sequences of refined meshes. Hierarchical basis functions can be constructed in a straightforward fashion on such sequences of nested meshes. The HBMG iteration itself is just a block symmetric GauSeidel iteration applied to the stiffness matrix represented in the hierarchical basis. Because the stiffness matrix is less sparse than when the standard nodal basis functions are used, the iteration is carried out by forming the hierarchical basis stiffness matrix only implicitly. The resulting algorithm is strongly connected to the classical multigrid Vcycle, except that only a subset of the unknowns on each level is smoothed during the relaxation steps [BDY88]. In recent years, we have generalized such bases to completely unstructured meshes, not just those arising from some refinement process. This is done by recognizing the strong connection between the Hierarchical Basis Multigrid Method and an