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216
Wavelet and Multiscale Methods for Operator Equations
 Acta Numerica
, 1997
"... this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of th ..."
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Cited by 171 (40 self)
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this paper is to highlight some of the underlying driving analytical mechanisms. The price of a powerful tool is the effort to construct and understand it. Its successful application hinges on the realization of a number of requirements. Some space has to be reserved for a clear identification of these requirements as well as for their realization. This is also particularly important for understanding the severe obstructions, that keep us at present from readily materializing all the principally promising perspectives.
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...
Stability of Multiscale Transformations
 J. Fourier Anal. Appl
, 1996
"... After briefly reviewing the interrelation between Rieszbases, biorthogonality and a certain stability notion for multiscale basis transformations we establish a basic stability criterion for a general Hilbert space setting. An important tool in this context is a strengthened Cauchy inequality. It i ..."
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Cited by 86 (22 self)
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After briefly reviewing the interrelation between Rieszbases, biorthogonality and a certain stability notion for multiscale basis transformations we establish a basic stability criterion for a general Hilbert space setting. An important tool in this context is a strengthened Cauchy inequality. It is based on direct and inverse estimates for a certain scale of spaces induced by the underlying multiresolution sequence. Furthermore, we highlight some properties of these spaces pertaining to duality, interpolation, and applications to norm equivalences for Sobolev spaces. AMS Subject Classification: 41A17, 41A65, 46A35, 46B70, 46E35 Key Words: Riesz bases, biorthogonality, stability, projectors, interpolation theory, Kmethod, duality, Jackson, Bernstein inequalities 1 Background and Motivation A standard framework for approximately recapturing a function v in some infinite dimensional separable Hilbert space V , say, either from explicitly given data or as a solution of an operator equ...
Biorthogonal SplineWavelets on the Interval  Stability and Moment Conditions
 Appl. Comp. Harm. Anal
, 1997
"... This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal Bsplines and co ..."
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Cited by 85 (46 self)
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This paper is concerned with the construction of biorthogonal multiresolution analyses on [0; 1] such that the corresponding wavelets realize any desired order of moment conditions throughout the interval. Our starting point is the family of biorthogonal pairs consisting of cardinal Bsplines and compactly supported dual generators on IR developed by Cohen, Daubechies and Feauveau. In contrast to previous investigations we preserve the full degree of polynomial reproduction also for the dual multiresolution and prove in general that the corresponding modifications of dual generators near the end points of the interval still permit the biorthogonalization of the resulting bases. The subsequent construction of compactly supported biorthogonal wavelets is based on the concept of stable completions. As a first step we derive an initial decomposition of the spline spaces where the complement spaces between two successive levels are spanned by compactly supported splines which form uniformly...
Domain Decomposition Algorithms With Small Overlap
, 1994
"... Numerical experiments have shown that twolevel Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear ..."
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Cited by 82 (11 self)
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Numerical experiments have shown that twolevel Schwarz methods often perform very well even if the overlap between neighboring subregions is quite small. This is true to an even greater extent for a related algorithm, due to Barry Smith, where a Schwarz algorithm is applied to the reduced linear system of equations that remains after that the variables interior to the subregions have been eliminated. In this paper, a supporting theory is developed.
UG  A Flexible Software Toolbox For Solving Partial Differential Equations
 COMPUTING AND VISUALIZATION IN SCIENCE
, 1997
"... Over the past two decades, some very efficient techniques for the numerical solution of partial differential equations have been developed. We are especially interested in adaptive local grid refinement on unstructured meshes, multigrid solvers and parallelization techniques. Up to now, these innova ..."
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Cited by 81 (20 self)
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Over the past two decades, some very efficient techniques for the numerical solution of partial differential equations have been developed. We are especially interested in adaptive local grid refinement on unstructured meshes, multigrid solvers and parallelization techniques. Up to now, these innovative techniques have been implemented mostly in university research codes and only very few commercial codes use them. There are two reasons for this. Firstly, the multigrid solution and adaptive refinement for many engineering applications are still a topic of active research and cannot be considered to be mature enough for routine application. Secondly, the implementation of all these techniques in a code with sufficient generality requires a lot of time and knowhow in different fields. UG (abbreviation for Unstructured Grids) has been designed to overcome these problems. It provides very general tools for the generation and manipulation of unstructured meshes in two and three space dime...
Multilevel Schwarz Methods For Elliptic Problems With Discontinuous Coefficients In Three Dimensions
 NUMER. MATH
, 1994
"... Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate fo ..."
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Cited by 54 (13 self)
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Multilevel Schwarz methods are developed for a conforming finite element approximation of second order elliptic problems. We focus on problems in three dimensions with possibly large jumps in the coefficients across the interface separating the subregions. We establish a condition number estimate for the iterative operator, which is independent of the coefficients, and grows at most as the square of the number of levels. We also characterize a class of distributions of the coefficients, called quasimonotone, for which the weighted L²projection is stable and for which we can use the standard piecewise linear functions as a coarse space. In this case, we obtain optimal methods, i.e. bounds which are independent of the number of levels and subregions. We also design and analyze multilevel methods with new coarse spaces given by simple explicit formulas. We consider nonuniform meshes and conclude by an analysis of multilevel iterative substructuring methods.
Combinatorial preconditioners for sparse, symmetric, diagonally dominant linear systems
, 1996
"... ..."
New Estimates for Multilevel Algorithms Including the VCycle
 MATH. COMP
, 1993
"... The purpose of this paper is to provide new estimates for certain multilevel algorithms. In particular, we are concerned with the simple additive multilevel algorithm given in [10] and the standard Vcycle algorithm with one smoothing step per grid. We shall prove that these algorithms have a unifo ..."
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Cited by 49 (5 self)
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The purpose of this paper is to provide new estimates for certain multilevel algorithms. In particular, we are concerned with the simple additive multilevel algorithm given in [10] and the standard Vcycle algorithm with one smoothing step per grid. We shall prove that these algorithms have a uniform reduction per iteration independent of the mesh sizes and number of levels even on nonconvex domains which do not provide full elliptic regularity. For example, the theory applies to the standard multigrid Vcycle on the Lshaped domain or a domain with a crack and yields a uniform convergence rate. We also prove uniform convergence rates for the multigrid Vcycle for problems with nonuniformly refined meshes. Finally, we give a new multigrid approach for problems on domains with curved boundaries and prove a uniform rate of convergence for the corresponding multigrid Vcycle algorithms.
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 ..."
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Cited by 48 (5 self)
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. 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, ...