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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
Adaptive Multilevel Methods in Three Space Dimensions
 INT. J. NUMER. METHODS ENG
, 1993
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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 51 (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, ...
Sparse grids and related approximation schemes for higher dimensional problems
"... The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach ..."
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Cited by 46 (12 self)
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The efficient numerical treatment of highdimensional problems is hampered by the curse of dimensionality. We review approximation techniques which overcome this problem to some extent. Here, we focus on methods stemming from Kolmogorov’s theorem, the ANOVA decomposition and the sparse grid approach and discuss their prerequisites and properties. Moreover, we present energynorm based sparse grids and demonstrate that, for functions with bounded mixed derivatives on the unit hypercube, the associated approximation rate in terms of the involved degrees of freedom shows no dependence on the dimension at all, neither in the approximation order nor in the order constant.
Multigrid method for H(DIV) in three dimensions
 ETNA
, 1997
"... . We are concerned with the design and analysis of a multigrid algorithm for H(div; ## elliptic linear variational problems. The discretization is based on H(div; conforming RaviartThomas elements. A thorough examination of the relevant bilinear form reveals that a separate treatment of vector ..."
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Cited by 33 (4 self)
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. We are concerned with the design and analysis of a multigrid algorithm for H(div; ## elliptic linear variational problems. The discretization is based on H(div; conforming RaviartThomas elements. A thorough examination of the relevant bilinear form reveals that a separate treatment of vector fields in the kernel of the divergence operator and its complement is paramount. We exploit the representation of discrete solenoidal vector fields as curls of finite element functions in socalled Nedelec spaces. It turns out that a combined nodal multilevel decomposition of both the RaviartThomas and Nedelec finite element spaces provides the foundation for a viable multigrid method. Its GauSeidel smoother involves an extra stage where solenoidal error components are tackled. By means of elaborate duality techniques we can show the asymptotic optimality in the case of uniform refinement. Numerical experiments confirm that the typical multigrid efficiency is actually achieved for model...
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.
SPARSE GRIDS FOR THE SCHRÖDINGER EQUATION
, 2005
"... We present a sparse grid/hyperbolic cross discretization for manyparticle problems. It involves the tensor product of a oneparticle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, differ ..."
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Cited by 16 (4 self)
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We present a sparse grid/hyperbolic cross discretization for manyparticle problems. It involves the tensor product of a oneparticle multilevel basis. Subsequent truncation of the associated series expansion then results in a sparse grid discretization. Here, depending on the norms involved, different variants of sparse grid techniques for manyparticle spaces can be derived that, in the best case, result in complexities and error estimates which are independent of the number of particles. Furthermore we introduce an additional constraint which gives antisymmetric sparse grids which are suited to fermionic systems. We apply the antisymmetric sparse grid discretization to the electronic Schrödinger equation and compare costs, accuracy, convergence rates and scalability with respect to the number of electrons present in the system.
Multigrid Preconditioning and Toeplitz Matrices
, 1999
"... In this paper we discuss Multigrid methods for Toeplitz matrices. Then the restriction and prolongation operator can be seen as projected Toeplitz matrices. Because of the intimate connection between such matrices and trigonometric series we can express the Multigrid algorithm in terms of the underl ..."
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Cited by 16 (1 self)
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In this paper we discuss Multigrid methods for Toeplitz matrices. Then the restriction and prolongation operator can be seen as projected Toeplitz matrices. Because of the intimate connection between such matrices and trigonometric series we can express the Multigrid algorithm in terms of the underlying functions with special zeroes. This shows how to choose the prolongation/restriction operator in order to get fast convergence. This approach allows Multigrid methods for general Toeplitz systems as long as we have some information on the underlying function. Furthermore, we can define projections not only on problems of half size n=2, but on every size n=m for a m 2 N . We can apply the derived method also to the constantcoefficient case for PDE on simple regions, e.g. in connection with the Helmholtz equation or ConvectionDiffusion equation.
Parallel geometric multigrid
 Numerical Solution of Partial Differential Equations on Parallel Computers, volume 51 of LNCSE, chapter 5
, 2005
"... Summary. Multigrid methods are among the fastest numerical algorithms for the solution of large sparse systems of linear equations. While these algorithms exhibit asymptotically optimal computational complexity, their efficient parallelisation is hampered by the poor computationtocommunication rat ..."
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Cited by 15 (6 self)
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Summary. Multigrid methods are among the fastest numerical algorithms for the solution of large sparse systems of linear equations. While these algorithms exhibit asymptotically optimal computational complexity, their efficient parallelisation is hampered by the poor computationtocommunication ratio on the coarse grids. Our contribution discusses parallelisation techniques for geometric multigrid methods. It covers both theoretical approaches as well as practical implementation issues that may guide code development. 1
Overlapping Schwarz Methods For Vector Valued Elliptic Problems In Three Dimensions
 IMA Volumes in Mathematics and its Applications
, 1997
"... . This paper is intended as a survey of current results on algorithmic and theoretical aspects of overlapping Schwarz methods for discrete H(curl; and H(div; \Omega\Gamma40199/ ic problems set in suitable finite element spaces. The emphasis is on a unified framework for the motivation and theoret ..."
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Cited by 13 (1 self)
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. This paper is intended as a survey of current results on algorithmic and theoretical aspects of overlapping Schwarz methods for discrete H(curl; and H(div; \Omega\Gamma40199/ ic problems set in suitable finite element spaces. The emphasis is on a unified framework for the motivation and theoretical study of the various approaches developed in recent years. Generalized Helmholtz decompositions  orthogonal decompositions into the null space of the relevant differential operator and its complement  are crucial in our considerations. It turns out that the decompositions the Schwarz methods are based upon have to be designed separately for both components. In the case of the null space, the construction has to rely on liftings into spaces of discrete potentials. Taking the cue from wellknown Schwarz schemes for second order elliptic problems, we devise uniformly stable splittings of both parts of the Helmholtz decomposition. They immediately give rise to powerful preconditioners...