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594
Reduced basis approximation and a posteriori error estimation for parametrized partial differential equations. Version 1.0, Copyright MIT
, 2006
"... reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations. The essential ingredients are (primaldual) Galerkin projection onto a lowdimensional space associated with a smooth “parametric ..."
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Cited by 205 (38 self)
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reduced basis approximation and a posteriori error estimation for linear functional outputs of affinely parametrized elliptic coercive partial differential equations. The essential ingredients are (primaldual) Galerkin projection onto a lowdimensional space associated with a smooth “parametric manifold”—dimension reduction; efficient and effective greedy sampling methods for identification of optimal and numerically stable approximations—rapid convergence; a posteriori error estimation procedures—rigorous and sharp bounds for the linearfunctional outputs of interest; and OfflineOnline computational decomposition strategies—minimum marginal cost for high performance in the realtime/embedded (e.g., parameterestimation, con
Grid adaptation for functional outputs: application to twodimensional inviscid flows
 J. Comput. Phys
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Quasioptimal convergence rate for an adaptive finite element method
, 2007
"... We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As it is customary in practice, AFEM marks exclusively according to the error estimator and performs a minimal element refineme ..."
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Cited by 102 (15 self)
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We analyze the simplest and most standard adaptive finite element method (AFEM), with any polynomial degree, for general second order linear, symmetric elliptic operators. As it is customary in practice, AFEM marks exclusively according to the error estimator and performs a minimal element refinement without the interior node property. We prove that AFEM is a contraction for the sum of energy error and scaled error estimator, between two consecutive adaptive loops. This geometric decay is instrumental to derive optimal cardinality of AFEM. We show that AFEM yields a decay rate of energy error plus oscillation in terms of number of degrees of freedom as dictated by the best approximation for this combined nonlinear quantity.
The heterogeneous multiscale method: A review
 COMMUN. COMPUT. PHYS
, 2007
"... This paper gives a systematic introduction to HMM, the heterogeneous multiscale method, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem. This is illustrated by examples from several applic ..."
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Cited by 100 (5 self)
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This paper gives a systematic introduction to HMM, the heterogeneous multiscale method, including the fundamental design principles behind the HMM philosophy and the main obstacles that have to be overcome when using HMM for a particular problem. This is illustrated by examples from several application areas, including complex fluids, microfluidics, solids, interface problems, stochastic problems, and statistically selfsimilar problems. Emphasis is given to the technical tools, such as the various constrained molecular dynamics, that have been developed, in order to apply HMM to these problems. Examples of mathematical results on the error analysis of HMM are presented. The paper ends with a discussion on some of
Optimality of a standard adaptive finite element method
"... In this paper, an adaptive ¯nite element method is constructed
for solving elliptic equations that has optimal computational complexity.
Whenever for some s > 0, the solution can be approximated to accuracy
O(n¡s) in energy norm by a continuous piecewise linear function on some
partition with n t ..."
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Cited by 91 (6 self)
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In this paper, an adaptive ¯nite element method is constructed
for solving elliptic equations that has optimal computational complexity.
Whenever for some s > 0, the solution can be approximated to accuracy
O(n¡s) in energy norm by a continuous piecewise linear function on some
partition with n triangles, and one knows how to approximate the righthand
side in the dual norm with the same rate with piecewise constants, then
the adaptive method produces approximations that converge with this rate,
taking a number of operations that is of the order of the number of triangles
in the output partition. The method is similar in spirit to that from [SINUM,
38 (2000), pp.466{488] by Morin, Nochetto, and Siebert, and so in particular
it does not rely on a recurrent coarsening of the partitions. Although the
Poisson equation in two dimensions with piecewise linear approximation is
considered, it can be expected that the results generalize in several respects.
GoalOriented Error Estimation and Adaptivity for the Finite Element Method
 Comput. Math. Appl
, 1999
"... this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These socalled quantities of interest are characterized by linear functionals on ..."
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Cited by 75 (9 self)
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this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These socalled quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs. We present here the theory with respect to a class of elliptic boundaryvalue problems, and in particular, show how to obtain accurate estimates as well as upper and lowerbounds on the error. We also study the new concept of goaloriented adaptivity, which embodies mesh adaptation procedures designed to control error in specific quantities. Numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates.
Convergence of adaptive finite element methods
 SIAM Review
"... Abstract. We prove convergence of adaptive finite element methods (AFEM) for general (nonsymmetric) second order linear elliptic PDE, thereby extending the result of Morin et al [6, 7]. The proof relies on quasiorthogonality, which accounts for the bilinear form not being a scalar product, together ..."
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Cited by 72 (6 self)
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Abstract. We prove convergence of adaptive finite element methods (AFEM) for general (nonsymmetric) second order linear elliptic PDE, thereby extending the result of Morin et al [6, 7]. The proof relies on quasiorthogonality, which accounts for the bilinear form not being a scalar product, together with novel error and oscillation reduction estimates, which now do not decouple. We show that AFEM is a contraction for the sum of energy error plus oscillation. Numerical experiments, including oscillatory coefficients and convectiondiffusion PDE, illustrate the theory and yield optimal meshes.
Averaging techniques yield reliable a posteriori finite element error control for obstacle problems
, 2001
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A Posteriori Finite Element Bounds for LinearFunctional Outputs of Elliptic Partial Differential Equations
 Computer Methods in Applied Mechanics and Engineering
, 1997
"... We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order elliptic linear partial differential equations in two space dimensions. The method is base ..."
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Cited by 63 (9 self)
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We present a domain decomposition finite element technique for efficiently generating lower and upper bounds to outputs which are linear functionals of the solutions to symmetric or nonsymmetric second order elliptic linear partial differential equations in two space dimensions. The method is based upon the construction of an augmented Lagrangian, in which the objective is a quadratic "energy" reformulation of the desired output, and the constraints are the finite element equilibrium equations and intersubdomain continuity requirements. The bounds on the output for a suitably fine "truthmesh" discretization are then derived by appealing to a dual maxmin relaxation evaluated for optimally chosen adjoint and hybridflux candidate Lagrange multipliers generated by a Kelement coarser "workingmesh" approximation. Independent of the form of the original partial differential equation, the computation on the truth mesh is reduced to K decoupled subdomainlocal, symmetric Neumann pro...
Estimation of Local Modeling Error and GoalOriented Adaptive Modeling of Heterogeneous Materials; Part I : Error Estimates and Adaptive Algorithms
 of Heterogeneous Materials; Part I : Error Estimates and Adaptive
"... . A theory of a posteriori estimation of modeling errors in local quantities of interest in the analysis of heterogeneous elastic solids is presented. These quantities may, for example, represent averaged stresses on the surface of inclusions or mollications of pointwise stresses or displacements, o ..."
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Cited by 54 (6 self)
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. A theory of a posteriori estimation of modeling errors in local quantities of interest in the analysis of heterogeneous elastic solids is presented. These quantities may, for example, represent averaged stresses on the surface of inclusions or mollications of pointwise stresses or displacements, or, in general, local features of the \nescale" solution characterized by continuous linear functionals. These estimators are used to construct goaloriented adaptive procedures in which models of the microstructure are adapted so as to deliver local features to a preset level of accuracy. Algorithms for implementing these procedures are discussed and preliminary numerical results are given. 1 Introduction The idea of automatically adapting characteristics of mathematical and computational models of heterogeneous media so as to obtain results of a specied level of accuracy was advanced in recent work on hierarchical modeling [11, 7]. In these papers, a posteriori bounds on the error in s...