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739
Sickel: Optimal approximation of elliptic problems by linear and nonlinear mappings III
 Triebel, Function Spaces, Entropy Numbers, Differential Operators
, 1996
"... We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: a) linear mappings of rank n; b) nterm approximation with respect to a Riesz basis; c) approximation based on linear information about the right hand side f; d) continuous mappings. We co ..."
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Cited by 135 (28 self)
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We study the optimal approximation of the solution of an operator equation A(u) = f by four types of mappings: a) linear mappings of rank n; b) nterm approximation with respect to a Riesz basis; c) approximation based on linear information about the right hand side f; d) continuous mappings. We consider worst case errors, where f is an element of the unit ball of a Sobolev or Besov space Br q(Lp(Ω)) and Ω ⊂ Rd is a bounded Lipschitz domain; the error is always measured in the Hsnorm. The respective widths are the linear widths (or approximation numbers), the nonlinear widths, the Gelfand widths, and the manifold widths. As a technical tool, we also study the Bernstein numbers. Our main results are the following. If p ≥ 2 then the order of convergence is the same for all four classes of approximations. In particular, the best linear approximations are of the same order as the best nonlinear ones. The best linear approximation can be quite difficult to realize as a numerical algorithm since the optimal Galerkin space usually depends on the operator and of the shape of the domain Ω. For p < 2 there is a difference, nonlinear approximations are better than linear ones. However, in this case, it turns out that linear information about the right hand side f is again optimal. Our main theoretical tool is the best nterm approximation with respect to an optimal Riesz basis and related nonlinear widths. These general results are used to study the Poisson equation in a polygonal domain. It turns out that best nterm wavelet approximation is (almost) optimal. The main results of
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 101 (12 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.
Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes
 SIAM J. Numer. Anal
, 2007
"... The stability and convergence properties of the mimetic finite difference method for diffusiontype problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved. 1 ..."
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Cited by 95 (20 self)
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The stability and convergence properties of the mimetic finite difference method for diffusiontype problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved. 1
An A Priori Error Analysis Of The Local Discontinuous Galerkin Method For Elliptic Problems
, 2000
"... . In this paper, we present the first a priori error analysis for the Local Discontinuous Galerkin method for a model elliptic problem. For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the L 2 norm of the gradient and ..."
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Cited by 94 (24 self)
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. In this paper, we present the first a priori error analysis for the Local Discontinuous Galerkin method for a model elliptic problem. For arbitrary meshes with hanging nodes and elements of various shapes, we show that, for stabilization parameters of order one, the L 2 norm of the gradient and the L 2 norm of the potential are of order k and k + 1=2, respectively, when polynomials of total degree at least k are used; if stabilization parameters of order h \Gamma1 are taken, the order of convergence of the potential increases to k + 1. The optimality of these theoretical results are tested in a series of numerical experiments on two dimensional domains. Key words. Finite elements, discontinuous Galerkin methods, elliptic problems AMS subject classifications. 65N30 1. Introduction. In this paper, we present the first a priori error analysis of the Local Discontinuous Galerkin (LDG) method for the following classical model elliptic problem: \Gamma\Deltau = f in\Omega ; u ...
A mixed multiscale finite element method for elliptic problems with oscillating coefficients
 MATH. COMP
, 2002
"... The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the largescale structure of the solutions without resolving all the finescale structures. Motivated by the numerical simulation of flow transport in highly h ..."
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Cited by 80 (11 self)
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The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the largescale structure of the solutions without resolving all the finescale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an oversampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random lognormal relative permeability to demonstrate the efficiency and accuracy of the proposed method.
Finite Element Methods and Their Convergence for Elliptic and Parabolic Interface Problems
 Numer. Math
, 1996
"... In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in twodimensional convex polygonal domains. Nearly the same optimal L 2 norm and energynorm error estimates as for regular problems are obtained when the interfaces are of ar ..."
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Cited by 78 (11 self)
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In this paper, we consider the finite element methods for solving second order elliptic and parabolic interface problems in twodimensional convex polygonal domains. Nearly the same optimal L 2 norm and energynorm error estimates as for regular problems are obtained when the interfaces are of arbitrary shape but are smooth, though the regularities of the solutions are low on the whole domain. The assumptions on the finite element triangulation are reasonable and practical. Mathematics Subject Classification (1991): 65N30, 65F10. A running title: Finite element methods for interface problems. Correspondence to: Dr. Jun Zou Email: zou@math.cuhk.edu.hk Fax: (852) 2603 5154 1 Institute of Mathematics, Academia Sinica, Beijing 100080, P.R. China. Email: zmchen@math03.math.ac.cn. The work of this author was partially supported by China National Natural Science Foundation. 2 Department of Mathematics, the Chinese University of Hong Kong, Shatin, N.T., Hong Kong. Email: zou@math.cuhk....
Mixed finite element methods on nonmatching multiblock grids
 SIAM J. Numer. Anal
, 2000
"... Abstract. We consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids. A mortar finite element space is introduced on the nonmatching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of flux ..."
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Cited by 71 (32 self)
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Abstract. We consider mixed finite element methods for second order elliptic equations on nonmatching multiblock grids. A mortar finite element space is introduced on the nonmatching interfaces. We approximate in this mortar space the trace of the solution, and we impose weakly a continuity of flux condition. A standard mixed finite element method is used within the blocks. Optimal order convergence is shown for both the solution and its flux. Moreover, at certain discrete points, superconvergence is obtained for the solution and also for the flux in special cases. Computational results using an efficient parallel domain decomposition algorithm are presented in confirmation of the theory.
On traces for functional spaces related to Maxwell's Equations  Part I: An integration...
, 1999
"... Part I. The aim of this paper is to study the tangential trace and tangential components of fields which belong to the space H(curl ;\Omega\Gamma2 when\Omega is a polyhedron with Lipschitz continuous boundary. The appropriate functional setting is developed in order to suitably define these traces ..."
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Cited by 67 (5 self)
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Part I. The aim of this paper is to study the tangential trace and tangential components of fields which belong to the space H(curl ;\Omega\Gamma2 when\Omega is a polyhedron with Lipschitz continuous boundary. The appropriate functional setting is developed in order to suitably define these traces on the whole boundary and on a part of it (for partially vanishing fields and general ones.) In both cases it is possible to define ad hoc dualities among tangential trace and tangential components and the validity of two related integration by parts formulae is provided. Part II. Hodge decompositions of tangential vector fields defined on piecewise regular manifolds are provided. The first step is the study of L 2 tangential fields and then the attention is focused on some particular Sobolev spaces of order \Gamma1=2. In order to reach this goal, it is required to properly define the first order differential operators and to study their properties. When the manifold \Gamma is the boundar...
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 61 (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.
A Regularizing LevenbergMarquardt Scheme, With Applications To Inverse Groundwater Filtration Problems
 Inverse Problems
, 1997
"... . The first part of this paper studies a LevenbergMarquardt scheme for nonlinear inverse problems where the corresponding Lagrange (or regularization) parameter is chosen from an inexact Newton strategy. While the convergence analysis of standard implementations based on trust region strategies alw ..."
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Cited by 57 (2 self)
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. The first part of this paper studies a LevenbergMarquardt scheme for nonlinear inverse problems where the corresponding Lagrange (or regularization) parameter is chosen from an inexact Newton strategy. While the convergence analysis of standard implementations based on trust region strategies always requires the invertibility of the Fr'echet derivative of the nonlinear operator at the exact solution, the new LevenbergMarquardt scheme is suitable for illposed problems as long as the Taylor remainder is of second order in the interpolating metric between the range and domain topologies. Estimates of this type are established in the second part of the paper for illposed parameter identification problems arising in inverse groundwater hydrology. Both, transient and steady state data are investigated. Finally, the numerical performance of the new LevenbergMarquardt scheme is studied and compared to a usual implementation on a realistic but synthetic 2D model problem from the engineer...