Results 1  10
of
42
Adaptive Multilevel Methods in Three Space Dimensions
 INT. J. NUMER. METHODS ENG
, 1993
"... ..."
(Show Context)
An Algorithm for Coarsening Unstructured Meshes
 Numer. Math
, 1996
"... . We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the lin ..."
Abstract

Cited by 51 (5 self)
 Add to MetaCart
(Show Context)
. We develop and analyze a procedure for creating a hierarchical basis of continuous piecewise linear polynomials on an arbitrary, unstructured, nonuniform triangular mesh. Using these hierarchical basis functions, we are able to define and analyze corresponding iterative methods for solving the linear systems arising from finite element discretizations of elliptic partial differential equations. We show that such iterative methods perform as well as those developed for the usual case of structured, locally refined meshes. In particular, we show that the generalized condition numbers for such iterative methods are of order J 2 , where J is the number of hierarchical basis levels. Key words. Finite element, hierarchical basis, multigrid, unstructured mesh. AMS subject classifications. 65F10, 65N20 1. Introduction. Iterative methods using the hierarchical basis decomposition have proved to be among the most robust for solving broad classes of elliptic partial differential equations, ...
A Posteriori Error Estimates for Elliptic Problems in Two and Three Space Dimensions
 SIAM J. NUMER. ANAL
, 1993
"... Let u 2 H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error jj u \Gamma ~ u jj, measuring the (local) quality of ~ u, play a cruci ..."
Abstract

Cited by 50 (8 self)
 Add to MetaCart
Let u 2 H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error jj u \Gamma ~ u jj, measuring the (local) quality of ~ u, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A wellknown class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations.
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 ..."
Abstract

Cited by 50 (13 self)
 Add to MetaCart
(Show Context)
. 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...
Residual Type A Posteriori Error Estimates For Elliptic Obstacle Problems
 Numer. Math
"... . A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positiv ..."
Abstract

Cited by 35 (11 self)
 Add to MetaCart
(Show Context)
. A posteriori error estimators of residual type are derived for piecewise linear finite element approximations to elliptic obstacle problems. An instrumental ingredient is a new interpolation operator which requires minimal regularity, exhibits optimal approximation properties and preserves positivity. Both upper and lower bounds are proved and their optimality is explored with several examples. Sharp a priori bounds for the a posteriori estimators are given, and extensions of the results to double obstacle problems are briefly discussed. Key words. a posteriori error estimates, residual, elliptic obstacle, positivity preserving interpolation. 1991 Mathematics Subject Classification. 65N15, 65N30; 41A05, 41A29, 41A36 1 Introduction A posteriori error estimates are computable quantities in terms of the discrete solution and data, which are instrumental for adaptive mesh refinement (and coarsening), error control, and equidistribution of the computational effort. Since the seminal pa...
Multilevel methods for elliptic problems on domains not resolved by the coarse grid
 Contemporary Mathematics
, 1994
"... ..."
(Show Context)
Convergence rate analysis of an asynchronous space decomposition method for convex minimization
, 1998
"... Abstract. We analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive Banach space. This method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equatio ..."
Abstract

Cited by 28 (11 self)
 Add to MetaCart
(Show Context)
Abstract. We analyze the convergence rate of an asynchronous space decomposition method for constrained convex minimization in a reflexive Banach space. This method includes as special cases parallel domain decomposition methods and multigrid methods for solving elliptic partial differential equations. In particular, the method generalizes the additive Schwarz domain decomposition methods to allow for asynchronous updates. It also generalizes the BPX multigrid method to allow for use as solvers instead of as preconditioners, possibly with asynchronous updates, and is applicable to nonlinear problems. Applications to an overlapping domain decomposition for obstacle problems are also studied. The method of this work is also closely related to relaxation methods for nonlinear network flow. Accordingly, we specialize our convergence rate results to the above methods. The asynchronous method is implementable in a multiprocessor system, allowing for communication and computation delays among the processors. 1.
The Hierarchical Basis Multigrid Method And Incomplete LU Decomposition
 In Seventh International Symposium on Domain Decomposition Methods for Partial Differential Equations
, 1994
"... . A new multigrid or incomplete LU technique is developed in this paper for solving large sparse algebraic systems from discretizing partial differential equations. By exploring some deep connection between the hierarchical basis method and incomplete LU decomposition, the resulting algorithm can be ..."
Abstract

Cited by 28 (7 self)
 Add to MetaCart
(Show Context)
. A new multigrid or incomplete LU technique is developed in this paper for solving large sparse algebraic systems from discretizing partial differential equations. By exploring some deep connection between the hierarchical basis method and incomplete LU decomposition, the resulting algorithm can be effectively applied to problems discretized on completelyunstructured grids. Numerical experiments demonstrating the efficiency of the method are also reported. Key words. Finite element, hierarchical basis, multigrid, incomplete LU . AMS(MOS) subject classifications. 65F10, 65N20 1. Introduction. In this work, we explore the connection between the methods of sparse Gaussian elimination [8][13], incomplete LU (ILU) decomposition [9][10] and the hierarchical basis multigrid (HBMG) [16][4]. Hierarchical basis methods have proved to be one of the more robust classes of methods for solving broad classes of elliptic partial differential equations, especially the large systems arising in conju...
Rate of Convergence for some constraint decomposition methods for nonlinear variational inequalities
 NUMER. MATH. (2003) 93: 755–786
, 2003
"... ..."
Adaptive Multigrid Methods for Signorini's Problem in Linear Elasticity
 Computing and Visualization in Science
, 2001
"... this paper we use a direct approach as introduced in [23, 26]. Our algorithm does not involve any regularization or dual formulation and should be considered as a descent method rather than an active set strategy. The basic idea is to minimize the energy on suitably selected ddimensional subspaces, ..."
Abstract

Cited by 16 (5 self)
 Add to MetaCart
this paper we use a direct approach as introduced in [23, 26]. Our algorithm does not involve any regularization or dual formulation and should be considered as a descent method rather than an active set strategy. The basic idea is to minimize the energy on suitably selected ddimensional subspaces, where d  2, 3 is the dimension of the deformed body. In this way, we obtain nonlinear variants of successive subspace correction methods in the sense of Xu [34]. See e.g. [9, 30] for a similar approach to smooth nonlinear problems. Wellknown projected block GaufiSeidel relaxation is recovered by choosing the ddimensional subspaces spanned by the fine grid nodal basis functions associated with a fixed node. In order to increase convergence speed by better representation of the lowfrequency components of the error, we additionally minimize on subspaces spanned by functions with large support. The suitable selection of these coarse grid spaces is crucial for the efficiency of the resulting method. Our choice is based on sophisticated modifications of the multilevel nodal basis. Straightforward implementation of the resulting algorithm requires additional prolongations in order to check the constraints prescribed on the fine grid. As a consequence, the complexity of one iteration step is O(nl log n) for uniformly refined triangulations and might be even O(n) in the adaptive case. Optimal complexity of the multigrid Vcycle is recovered by approximating fine grid constraints on coarser grids using socalled monotone restrictions. This modification may slow down convergence, as long as the algebraic error is too large. In our numerical experiments we observed that initial iterates as provided by nested iteration are usually accurate enough to provide fast convergence through...