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37
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 172 (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.
Adaptive wavelet methods for elliptic operator equations— convergence rates
 Math. Comput
, 2001
"... Abstract. This paper is concerned with the construction and analysis of waveletbased adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u of the equation by a linear combination of N wavelets. Therefore, a benchmark for their performance ..."
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Cited by 109 (30 self)
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Abstract. This paper is concerned with the construction and analysis of waveletbased adaptive algorithms for the numerical solution of elliptic equations. These algorithms approximate the solution u of the equation by a linear combination of N wavelets. Therefore, a benchmark for their performance is provided by the rate of best approximation to u by an arbitrary linear combination of N wavelets (so called Nterm approximation), which would be obtained by keeping the N largest wavelet coefficients of the real solution (which of course is unknown). The main result of the paper is the construction of an adaptive scheme which produces an approximation to u with error O(N −s)in the energy norm, whenever such a rate is possible by Nterm approximation. The range of s>0 for which this holds is only limited by the approximation properties of the wavelets together with their ability to compress the elliptic operator. Moreover, it is shown that the number of arithmetic operations needed to compute the approximate solution stays proportional to N. The adaptive algorithm applies to a wide class of elliptic problems and wavelet bases. The analysis in this paper puts forward new techniques for treating elliptic problems as well as the linear systems of equations that arise from the wavelet discretization. 1.
Wavelets on Manifolds I: Construction and Domain Decomposition
 SIAM J. Math. Anal
, 1998
"... The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features ar ..."
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Cited by 80 (21 self)
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The potential of wavelets as a discretization tool for the numerical treatment of operator equations hinges on the validity of norm equivalences for Besov or Sobolev spaces in terms of weighted sequence norms of wavelet expansion coefficients and on certain cancellation properties. These features are crucial for the construction of optimal preconditioners, for matrix compression based on sparse representations of functions and operators as well as for the design and analysis of adaptive solvers. However, for realistic domain geometries the relevant properties of wavelet bases could so far only be realized to a limited extent. This paper is concerned with concepts that aim at expanding the applicability of wavelet schemes in this sense. The central issue is to construct wavelet bases with the desired properties on manifolds which can be represented as the disjoint union of smooth parametric images of the standard cube. The approach considered here is conceptually different though from o...
Stable Multiscale Bases and Local Error Estimation for Elliptic Problems
 Appl. Numer. Math
, 1996
"... This paper is concerned with the analysis of adaptive multiscale techniques for the solution of a wide class of elliptic operator equations covering, in principle, singular integral as well as partial differential operators. The central objective is to derive reliable and efficient aposteriori erro ..."
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Cited by 63 (33 self)
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This paper is concerned with the analysis of adaptive multiscale techniques for the solution of a wide class of elliptic operator equations covering, in principle, singular integral as well as partial differential operators. The central objective is to derive reliable and efficient aposteriori error estimators for Galerkin schemes which are based on stable multiscale bases. It is shown that the locality of corresponding multiresolution processes combined with certain norm equivalences involving weighted sequence norms leads to adaptive space refinement strategies which are guaranteed to converge in a wide range of cases, again including operators of negative order. Key words: Stable multiscale bases, norm equivalences, elliptic operator equations, Galerkin schemes, aposteriori error estimators, convergence of adaptive schemes AMS subject classification: 65N55, 65N30, 65N38, 65N12 1 Introduction The increasing importance of adaptive techniques in large scale computation is reflected...
Nonlinear Approximation and Adaptive Techniques for Solving Elliptic Operator Equations
, 1996
"... This survey article is concerned with two basic approximation concepts and their interrelation with the numerical solution of elliptic operator equations, namely nonlinear and adaptive approximation. On one hand, for nonlinear approximation based on wavelet expansions the best possible approxima ..."
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Cited by 30 (20 self)
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This survey article is concerned with two basic approximation concepts and their interrelation with the numerical solution of elliptic operator equations, namely nonlinear and adaptive approximation. On one hand, for nonlinear approximation based on wavelet expansions the best possible approximation rate, which a function can have for a given number of degrees of freedom, is characterized in terms of its regularity in a certain scale of Besov spaces. Therefore, after demonstrating the gain of nonlinear approximation over linear approximation measured in a Sobolev scale, we review some recent results on the Sobolev and Besov regularity of solutions to elliptic boundary value 1 problems. On the other hand, nonlinear approximation requires information that is generally not available in practice. Instead one has to resort to the concept of adaptive approximation. We briefly summarize some recent results on wavelet based adaptive schemes for elliptic operator equations. In co...
Residual Based A Posteriori Error Estimators For Eddy Current Computation
, 1999
"... We consider H(curl;\Omega\Gamma3932/608 problems that have been discretized by means of N'ed'elec's edge elements on tetrahedral meshes. Such problems occur in the numerical compuation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori err ..."
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Cited by 27 (6 self)
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We consider H(curl;\Omega\Gamma3932/608 problems that have been discretized by means of N'ed'elec's edge elements on tetrahedral meshes. Such problems occur in the numerical compuation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori error estimators to control adaptive refinement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtztype decomposition of the error into an irrotational part and a weakly solenoidal part.
A comparison of a posteriori error estimators for mixed finite element discretizations by raviartthomas elements
 MATH. COMP
, 1999
"... We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a resid ..."
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Cited by 27 (5 self)
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We consider mixed finite element discretizations of linear second order elliptic boundary value problems with respect to an adaptively generated hierarchy of possibly highly nonuniform simplicial triangulations. In particular, we present and analyze four different kinds of error estimators: a residual based estimator, a hierarchical one, error estimators relying on the solution of local subproblems and on a superconvergence result, respectively. Finally, we examine the relationship between the presented error estimators and compare their local components.
Hierarchical A Posteriori Error Estimators For Mortar Finite Element Methods With Lagrange Multipliers
 SIAM J. Numer. Anal
"... . Hierarchical a posteriori error estimators are introduced and analyzed for mortar finite element methods. A weak continuity condition at the interfaces is enforced by means of Lagrange multipliers. The two proposed error estimators are based on a defect correction in higher order finite element sp ..."
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Cited by 22 (6 self)
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. Hierarchical a posteriori error estimators are introduced and analyzed for mortar finite element methods. A weak continuity condition at the interfaces is enforced by means of Lagrange multipliers. The two proposed error estimators are based on a defect correction in higher order finite element spaces and an adequate hierarchical twolevel splitting. The first provides upper and lower bounds for the discrete energy norm of the mortar finite element solution whereas the second also estimates the error for the Lagrange multiplier. It is shown that an appropriate measure for the nonconformity of the mortar finite element solution is the weighted L 2 norm of the jumps across the interfaces. Key words. mortar finite elements, Lagrange multiplier, mesh dependent norms, a posteriori error estimation, adaptive grid refinement AMS subject classifications. 65N15, 65N30, 65N50, 65N55 1. Introduction. In this paper, we consider a special nonoverlapping domain decomposition method for the di...
Adaptive Multilevel Methods for Edge Element Discretizations of Maxwell's Equations
, 1997
"... . The focus of this paper is on the efficient solution of boundary value problems involving the doublecurl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackl ..."
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Cited by 21 (10 self)
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. The focus of this paper is on the efficient solution of boundary value problems involving the doublecurl operator. Those arise in the computation of electromagnetic fields in various settings, for instance when solving the electric or magnetic wave equation with implicit timestepping, when tackling timeharmonic problems or in the context of eddycurrent computations. Their discretization is based on on N'ed'elec's H(curl;\Omega\Gamma7131/59948 edge elements on unstructured grids. In order to capture local effects and to guarantee a prescribed accuracy of the approximate solution adaptive refinement of the grid controlled by a posteriori error estimators is employed. The hierarchy of meshes created through adaptive refinement forms the foundation for the fast iterative solution of the resulting linear systems by a multigrid method. The guiding principle underlying the design of both the error estimators and the multigrid method is the separate treatment of the kernel of the cu...
Adaptive Wavelet Schemes for Elliptic Problems  Implementation and Numerical Experiments
 SIAM J. Scient. Comput
, 1999
"... Recently an adaptive wavelet scheme could be proved to be asymptotically optimal for a wide class of elliptic operator equations in the sense that the error achieved by an adaptive approximate solution stays proportional to the smallest possible error that can be realized by any linear combination o ..."
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Cited by 19 (11 self)
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Recently an adaptive wavelet scheme could be proved to be asymptotically optimal for a wide class of elliptic operator equations in the sense that the error achieved by an adaptive approximate solution stays proportional to the smallest possible error that can be realized by any linear combination of the corresponding number of wavelets. On one hand, the results are purely asymptotic. On the other hand, the analysis suggests new algorithmic ingredients for which no prototypes seem to exist yet. It is therefore the objective of this paper to develop suitable data structures for the new algorithmic components and to obtain a quantitative validation of the theoretical results. We briey review rst the main theoretical facts, give a detailed description of the algorithm, highlight the essential data structures and illustrate the results by one and two dimensional numerical examples.