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48
Adaptive Finite Element Methods for Elliptic Equations With NonSmooth Coefficients
, 2000
"... We consider a secondorder elliptic equation with discontinuous or anisotropic coefficients in a bounded two or three dimensional domain, and its finiteelement discretization. The aim of this paper is to prove some a priori and a posteriori error estimates in an appropriate norm, which are independ ..."
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Cited by 49 (1 self)
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We consider a secondorder elliptic equation with discontinuous or anisotropic coefficients in a bounded two or three dimensional domain, and its finiteelement discretization. The aim of this paper is to prove some a priori and a posteriori error estimates in an appropriate norm, which are independent of the variation of the coefficients.
A posteriori analysis of the finite element discretization of some parabolic equations
 MR2136996 RECONSTRUCTION FOR DISCRETE PARABOLIC PROBLEMS 1657
, 2005
"... Abstract. We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respe ..."
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Cited by 32 (5 self)
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Abstract. We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respect to both time and space approximations and proving their equivalence with the error, in order to work with adaptive time steps and finite element meshes. Résumé. Nous considérons la discrétisation d’équations paraboliques, soit linéaires soit semilinéaires, par un schéma d’Euler implicite en temps et par éléments finis en espace. L’idée de cet article est de construire des indicateurs d’erreur liés à l’approximation en temps et en espace et de prouver leur équivalence avec l’erreur, dans le but de travailler avec des pas de temps adaptatifs et des maillages d’éléments finis adaptés à la solution. 1.
Asymptotically exact a posteriori estimators for the pointwise gradient error on each element in irregular meshes. I. A smooth problem and globally quasiuniform meshes
 Math. Comp
"... Abstract. A class of a posteriori estimators is studied for the error in the maximumnorm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems. The meshes are unstructured and, in particular, it is not assumed that there ..."
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Cited by 28 (4 self)
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Abstract. A class of a posteriori estimators is studied for the error in the maximumnorm of the gradient on single elements when the finite element method is used to approximate solutions of second order elliptic problems. The meshes are unstructured and, in particular, it is not assumed that there are any known superconvergent points. The estimators are based on averaging operators which are approximate gradients, “recovered gradients”, which are then compared to the actual gradient of the approximation on each element. Conditions are given under which they are asympotically exact or equivalent estimators on each single element of the underlying meshes. Asymptotic exactness is accomplished by letting the approximate gradient operator average over domains that are large, in a controlled fashion to be detailed below, compared to the size of the elements. 1.
A unifying theory of a posteriori error control for nonconforming finite element methods
 Numer. Math
"... Abstract. Residualbased a posteriori error estimates were derived within one unifying framework for lowestorder conforming, nonconforming, and mixed finite element schemes in [C. Carstensen, Numerische Mathematik 100 (2005) 617637]. Therein, the key assumption is that the conforming firstorder f ..."
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Cited by 22 (8 self)
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Abstract. Residualbased a posteriori error estimates were derived within one unifying framework for lowestorder conforming, nonconforming, and mixed finite element schemes in [C. Carstensen, Numerische Mathematik 100 (2005) 617637]. Therein, the key assumption is that the conforming firstorder finite element space V c h annulates the linear and bounded residual ℓ written V c h ⊆ ker ℓ. That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that V c h 6 ⊂ ker ℓ. The present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates, also for mortar and discontinuous Galerkin methods. The key assumption is the existence of some bounded linear operator Π: V c h → V nc h with some elementary properties. It is conjectured that the more general hypothesis (H1)(H3) can be established for all known NCFEMs. Applications on various nonstandard finite element schemes for the Laplace, Stokes, and NavierLame ́ equations illustrate the presented unifying theory of a posteriori error control for nonconforming finite element methods. 1. Unified Mixed Approach to Error Control Suppose that the primal variable u ∈ V (e.g., the displacement field) is accompanied by a dual variable p ∈ L (e.g., the flux or stress field). Typically L is some Lebesgue and V is some Sobolev space; suppose throughout this paper that L and V are Hilbert spaces and X: = L × V. Given bounded bilinear forms (1.1) a: L × L → R and b: L × V → R and well established conditions on a and b [16, 20], the linear and bounded operator
Local Error Estimates and Adaptive Refinement for FirstOrder System Least Squares (FOSLS)
, 1997
"... We establish an aposteriori error estimate, with corresponding bounds, that is valid for any FOSLS L 2 minimization problem. Such estimates follow almost immediately from the FOSLS formulation, but they are usually difficult to establish for other methodologies. We present some numerical example ..."
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Cited by 18 (5 self)
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We establish an aposteriori error estimate, with corresponding bounds, that is valid for any FOSLS L 2 minimization problem. Such estimates follow almost immediately from the FOSLS formulation, but they are usually difficult to establish for other methodologies. We present some numerical examples to support our theoretical results. We also establish a local apriori lower error bound that is useful for indicating when refinement is necessary and for determining the initial grid. Finally, we obtain a sharp theoretical error estimate under certain assumptions on the refinement region and show how this provides the basis for an effective refinement strategy. The local apriori lower error bound and the sharp theoretical error estimate both appear to be unique to the leastsquares approach.
An adaptive finite element approximation of a variational model of brittle fracture
 SIAM J. NUMER. ANAL
, 2008
"... The energy of the Francfort–Marigo model of brittle fracture can be approximated, in the sense of Γconvergence, by the Ambrosio–Tortorelli functional. In this work we formulate and analyze two adaptive finite element algorithms for the computation of its (local) minimizers. For each algorithm we ..."
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Cited by 15 (3 self)
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The energy of the Francfort–Marigo model of brittle fracture can be approximated, in the sense of Γconvergence, by the Ambrosio–Tortorelli functional. In this work we formulate and analyze two adaptive finite element algorithms for the computation of its (local) minimizers. For each algorithm we combine a Newtontype method with residualdriven adaptive mesh refinement. We present two theoretical results which demonstrate convergence of our algorithms to local minimizers of the Ambrosio–Tortorelli functional.
Adaptive multiresolution analysis based on anisotropic triangulations, preprint, Laboratoire J.L.Lions, submitted 2008
"... A simple greedy refinement procedure for the generation of dataadapted triangulations is proposed and studied. Given a function f of two variables, the algorithm produces a hierarchy of triangulations (Dj)j≥0 and piecewise polynomial approximations of f on these triangulations. The refinement proce ..."
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Cited by 13 (5 self)
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A simple greedy refinement procedure for the generation of dataadapted triangulations is proposed and studied. Given a function f of two variables, the algorithm produces a hierarchy of triangulations (Dj)j≥0 and piecewise polynomial approximations of f on these triangulations. The refinement procedure consists in bisecting a triangle T in a direction which is chosen so as to minimize the local approximation error in some prescribed norm between f and its piecewise polynomial approximation after T is bisected. The hierarchical structure allows us to derive various approximation tools such as multiresolution analysis, wavelet bases, adaptive triangulations based either on greedy or optimal CART trees, as well as a simple encoding of the corresponding triangulations. We give a general proof of convergence in the L p norm of all these approximations. Numerical tests performed in the case of piecewise linear approximation of functions with analytic expressions or of numerical images illustrate the fact that the refinement procedure generates triangles with an optimal aspect ratio (which is dictated by the local Hessian of f in case of C 2 functions). 1
A posteriori error analysis of the fully discretized timedependent Stokes equations
"... The timedependent Stokes equations in two or threedimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indic ..."
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Cited by 11 (2 self)
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The timedependent Stokes equations in two or threedimensional bounded domains are discretized by the backward Euler scheme in time and finite elements in space. The error of this discretization is bounded globally from above and locally from below by the sum of two types of computable error indicators, the first one being linked to the time discretization and the second one to the space discretization.
Error indicators for the mortar finite element discretization of the Laplace equation
 Math. Comput
, 2000
"... Abstract. The mortar technique turns out to be well adapted to handle mesh adaptivity in finite elements, since it allows for working with nonnecessarily compatible discretizations on the elements of a nonconforming partition of the initial domain. The aim of this paper is to extend the numerical an ..."
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Cited by 11 (4 self)
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Abstract. The mortar technique turns out to be well adapted to handle mesh adaptivity in finite elements, since it allows for working with nonnecessarily compatible discretizations on the elements of a nonconforming partition of the initial domain. The aim of this paper is to extend the numerical analysis of residual error indicators to this type of methods for a model problem and to check their efficiency thanks to some numerical experiments. 1.