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62
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 220 (37 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.
Hierarchical Bases and the Finite Element Method
, 1997
"... CONTENTS 1 Introduction 1 2 Preliminaries 3 3 Fundamental TwoLevel Estimates 7 4 A Posteriori Error Estimates 16 5 TwoLevel Iterative Methods 23 6 Multilevel Cauchy Inequalities 30 7 Multilevel Iterative Methods 34 References 41 1. Introduction In this work we present a brief introduction to hie ..."
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Cited by 80 (4 self)
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CONTENTS 1 Introduction 1 2 Preliminaries 3 3 Fundamental TwoLevel Estimates 7 4 A Posteriori Error Estimates 16 5 TwoLevel Iterative Methods 23 6 Multilevel Cauchy Inequalities 30 7 Multilevel Iterative Methods 34 References 41 1. Introduction In this work we present a brief introduction to hierarchical bases, and the important part they play in contemporary finite element calculations. In particular, we examine their role in a posteriori error estimation, and in the Department of Mathematics, University of California at San Diego, La Jolla, CA 92093. The work of this author was supported by the Office of Naval Research under contract N0001489J1440. 2 Randolph E. Bank formulation of iterative methods for solving the large sparse sets of linear equations arising from the finite element discretization. Our goal is that the development should be largely selfcontained, but at the same time accessible and interest
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 76 (20 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.
Adaptive Multilevel Methods in Three Space Dimensions
 INT. J. NUMER. METHODS ENG
, 1993
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The cascadic multigrid method for elliptic problems
, 1996
"... The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade principle view, a basic iteration method without ..."
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Cited by 54 (5 self)
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The paper deals with certain adaptive multilevel methods at the confluence of nested multigrid methods and iterative methods based on the cascade principle of [10]. From the multigrid point of view, no correction cycles are needed; from the cascade principle view, a basic iteration method without any preconditioner is used at successive refinement levels. For a prescribed error tolerance on the final level, more iterations must be spent on coarser grids in order to allow for less iterations on finer grids. A first candidate of such a cascadic multigrid method was the recently suggested cascadic conjugate gradient method of [9], in short CCG method, which used the CG method as basic iteration method on each level. In [18] it has been proven, that the CCG method is accurate with optimal complexity for elliptic problems in 2D and quasiuniform triangulations. The present paper simplifies that theory and extends it to more general basic iteration methods like the traditional multigrid smoothers. Moreover, an adaptive control strategy for the number of iterations on successive refinement levels for possibly highly nonuniform grids is worked out on the basis of a posteriori estimates. Numerical tests confirm the efficiency and robustness of the cascadic multigrid method.
Multilevel Algorithms Considered As Iterative Methods On Semidefinite Systems
 SIAM J. Sci. Comput
"... . For the representation of piecewise dlinear functions, instead of the usual finite element basis, we introduce a generating system that contains the nodal basis functions of the finest level and of all coarser levels of discretization. This approach enables us to work directly with multilevel dec ..."
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Cited by 43 (10 self)
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. For the representation of piecewise dlinear functions, instead of the usual finite element basis, we introduce a generating system that contains the nodal basis functions of the finest level and of all coarser levels of discretization. This approach enables us to work directly with multilevel decompositions of a function. For a partial differential equation, the Galerkin scheme based on this generating system results in a semidefinite matrix equation that has in the 1D case only about twice, in the 2D case about 4/3 times, and in the 3D case about 8/7 times, as many unknowns as the usual system. Furthermore, the semidefinite system possesses not just one, but many solutions. However, the unique solution of the usual definite finite element problem can be easily computed from every solution of the semidefinite problem. We show that modern multilevel algorithms can be considered as standard iterative methods over the semidefinite system. The conjugate gradient method (CG) for the semi...
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 42 (6 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.
Additive MultilevelPreconditioners Based On Bilinear Interpolation, MatrixDependent Geometric Coarsening And AlgebraicMultigrid Coarsening For Second Order Elliptic PDEs
 APPL. NUMER. MATH
, 1997
"... In this paper, we study additive multilevel preconditioners based on bilinear interpolation, matrixdependent interpolations and the algebraic multigrid approach. We consider 2nd order elliptic problems, i.e. strong elliptic ones, singular perturbation problems and problems with locally strongly var ..."
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Cited by 30 (12 self)
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In this paper, we study additive multilevel preconditioners based on bilinear interpolation, matrixdependent interpolations and the algebraic multigrid approach. We consider 2nd order elliptic problems, i.e. strong elliptic ones, singular perturbation problems and problems with locally strongly varying or discontinuous coefficient functions. We report on the results of our numerical experiments which show that especially the algebraic multigrid based method is mostly robust also in the additive case.
Multilevel methods for elliptic problems on domains not resolved by the coarse grid
 Contemporary Mathematics
, 1994
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Stabilizing the Hierarchical Basis by Approximate Wavelets II: Implementation and Numerical Results
 I: Theory, Numer. Linear Alg. Appl., 4 Number
, 1998
"... This paper is the second part of a work on stabilizing the classical hierarchical basis HB by using waveletlike basis functions. Implementation techniques are of major concern for the multilevel preconditioners proposed by the authors in the first part of the work, which deals with algorithms and t ..."
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Cited by 28 (3 self)
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This paper is the second part of a work on stabilizing the classical hierarchical basis HB by using waveletlike basis functions. Implementation techniques are of major concern for the multilevel preconditioners proposed by the authors in the first part of the work, which deals with algorithms and their mathematical theory. Numerical results are presented to confirm the theory established there. A comparison of the performance of a number of multilevel methods is conducted for elliptic problems of three space variables. Key words. hierarchical basis, multilevel methods, preconditioning, finite element elliptic equations, approximate wavelets AMS subject classifications. 65F10, 65N20, 65N30 PII. S1064827596300668 1.