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16
Multilevel Preconditioning
 Numer. Math
, 1992
"... This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding ..."
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Cited by 94 (18 self)
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This paper is concerned with multilevel techniques for preconditioning linear systems arising from Galerkin methods for elliptic boundary value problems. A general estimate is derived which is based on the characterization of Besov spaces in terms of weighted sequence norms related to corresponding multilevel expansions. The result brings out clearly how the various ingredients of a typical multilevel setting affect the growth rate of the condition numbers. In particular, our analysis indicates how to realize even uniformly bounded condition numbers. For example, the general results are used to show that the BramblePasciakXu preconditioner for piecewise linear finite elements gives rise to uniformly bounded condition numbers even when the refinements of the underlying triangulations are highly nonuniform. Furthermore, they are applied to a general multivariate setting of refinable shiftinvariant spaces, in particular, covering those induced by various types of wavelets. Key words: G...
Adaptive Multilevel Methods in Three Space Dimensions
 Int. J. Numer. Methods Eng
, 1993
"... this paper to collect wellknown results on 3D mesh refinement ..."
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Cited by 42 (6 self)
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this paper to collect wellknown results on 3D mesh refinement
Parallel Multigrid in an Adaptive PDE Solver Based on Hashing and SpaceFilling Curves
, 1997
"... this paper is organized as follows: In section 2 we discuss data structures for adaptive PDE solvers. Here, we suggest to use hash tables instead of the usually employed tree type data structures. Then, in section 3 we discuss the main features of the sequential adaptive multilevel solver. Section 4 ..."
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Cited by 39 (3 self)
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this paper is organized as follows: In section 2 we discuss data structures for adaptive PDE solvers. Here, we suggest to use hash tables instead of the usually employed tree type data structures. Then, in section 3 we discuss the main features of the sequential adaptive multilevel solver. Section 4 deals with the partitioning and distribution of adaptive grids with spacefilling curves and section 5 gives the main features of our new parallelized adaptive multilevel solver. In section 6 we present the results of numerical experiments on a parallel cluster computer with up to 64 processors. It is shown that our approach works nicely also for problems with severe singularities which result in locally refined meshes. Here, the work overhead for load balancing and data distribution remains only a small fraction of the overall work load. 2. DATA STRUCTURES FOR ADAPTIVE PDE SOLVERS 2.1. Adaptive Cycle
Hierarchical Bases
 In ICIAM91. SIAM
, 1992
"... . The solution of the large linear systems arising from finite element discretizations of elliptic boundary value problems is a basic task in numerical analysis. For uniformly refined grids, multigrid methods are well established and very efficient methods to solve these problems. For adaptively gen ..."
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Cited by 30 (2 self)
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. The solution of the large linear systems arising from finite element discretizations of elliptic boundary value problems is a basic task in numerical analysis. For uniformly refined grids, multigrid methods are well established and very efficient methods to solve these problems. For adaptively generated, strongly nonuniform grids, the situation is more complicated, both with regard to convergence properties and with regard to the computational complexity of the single iteration step. The paper discusses a surprisingly simple approach which is very well suited to nonuniform grids, the hierarchical decomposition of finite element spaces. The method is based on an old idea from the theory of real functions, which is often used to produce fractal curves and surfaces. It is closely related to recent L 2 like decompositions. Key words. hierarchical bases, fast iterative solvers, finite elements AMS(MOS) subject classifications. 65N22, 65N30, 65N50, 65N55, 26A27 1. Introduction. Toda...
Adaptive Multilevel  Methods for Obstacle Problems
, 1992
"... We consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by ..."
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Cited by 26 (3 self)
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We consider the discretization of obstacle problems for second order elliptic differential operators by piecewise linear finite elements. Assuming that the discrete problems are reduced to a sequence of linear problems by suitable active set strategies, the linear problems are solved iteratively by preconditioned cgiterations. The proposed preconditioners are treated theoretically as abstract additive Schwarz methods and are implemented as truncated hierarchical basis preconditioners. To allow for local mesh refinement we derive semilocal and local a posteriori error estimates, providing lower and upper estimates for the discretization error. The theoretical results are illustrated by numerical computations.
HashStorage Techniques for Adaptive Multilevel Solvers and Their Domain Decomposition Parallelization
 Domain decomposition methods 10. The 10th int. conf., Boulder, volume 218 of Contemp. Math
, 1998
"... this article remain attractive even for such a code. ..."
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Cited by 18 (6 self)
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this article remain attractive even for such a code.
Comparison of Parallel Solvers for Nonlinear Elliptic Problems Based on Domain Decomposition Ideas
 PARALLEL COMPUTING
, 1995
"... In the present paper, the solution of nonlinear elliptic boundary value problems (b.v.p.) on parallel machines with Multiple Instruction Multiple Data (MIMD) architecture is discussed. Especially, we consider electromagnetic field problems the numerical solution of which is based on finite elem ..."
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Cited by 16 (5 self)
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In the present paper, the solution of nonlinear elliptic boundary value problems (b.v.p.) on parallel machines with Multiple Instruction Multiple Data (MIMD) architecture is discussed. Especially, we consider electromagnetic field problems the numerical solution of which is based on finite element discretizations and a nested Newton solver. For solving the linear systems of algebraic finite element equations in each Newton step, parallel conjugate gradient methods with a Domain Decomposition preconditioner (DD PCG) as well as parallelized global multigrid methods are applied. The implementation of the whole algorithm, i.e. the mesh generation, the generation of the finite element equations, the nested Newton algorithm, the DD PCG method and the global multigrid method, is based on a nonoverlapping DD data structure. The efficiency of the parallel DD PCG methods and the parallelized global multigrid methods, which are embedded in the nested Newton solver, are compared. Fur...
Data Structures And Concepts For Adaptive Finite Element Methods
, 1995
"... Zusammenfassung Data Structures and Concepts for Adaptive Finite Element Methods. The administration of strongly nonuniform, adaptively generated finite element meshes requires specialized techniques and data structures. A special data structure of this kind is described in this paper. It relies on ..."
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Cited by 11 (1 self)
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Zusammenfassung Data Structures and Concepts for Adaptive Finite Element Methods. The administration of strongly nonuniform, adaptively generated finite element meshes requires specialized techniques and data structures. A special data structure of this kind is described in this paper. It relies on points, edges and triangles as basic structures and is especially well suited for the realization of iterative solvers like the hierarchical basis or the multilevel nodal basis method. AMS Subject Classifications: 65N50, 65Y99, 65N30, 65N55, 65F10 Key words: Data structures, adaptive finite element methods. Datenstrukturen und Strategien fur adaptive Finite Elemente. Fur die Verwaltung von extrem nichtuniformen, adaptiv erzeugten Finite Element Gittern benotigt man spezielle Techniken und Datenstrukturen. Eine Datenstruktur dieser Art wird in diesem Artikel beschrieben. Basisstrukturen sind Punkte, Kanten und Dreiecke. Die Datenstruktur ist besonders zugeschnitten auf iterative Loser wie die hierarchische Basis oder die "multilevel nodal basis" Methode. 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 8 (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...
Hash Based Adaptive Parallel Multilevel Methods with SpaceFilling Curves
 NIC Series
, 2002
"... this paper a parallelisable and cheap method based on spacefilling curves is proposed. The partitioning is embedded into the parallel solution algorithm using multilevel iterative solvers and adaptive grid refinement. Numerical experiments on two massively parallel computers prove the efficienc ..."
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Cited by 8 (0 self)
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this paper a parallelisable and cheap method based on spacefilling curves is proposed. The partitioning is embedded into the parallel solution algorithm using multilevel iterative solvers and adaptive grid refinement. Numerical experiments on two massively parallel computers prove the efficiency of this approach