1 Introduction.
This work deals with an **a** **posteriori** **error** **estimator** for hermitian positive eigenvalue problems. Having **a** sharp **error** **estimator** for eigenvalues is, in general, **a** difficult task. Several works were proposed in the literature: in [2, 3] **error** estimators are proposed for the finite element discretization of eigenvalues of elliptic operators. In the reduced basis context for Stokes equations, an **a** **posteriori** **error** **estimator** based on Babuška stability theory [1] is proposed in [8]. **A** general formulation of **a** **posteriori** **error** bounds that can be applied to several situations is given in [5]. We also refer the reader to thesis manuscript [7] where **error** analysis is carried on for numerous problems: coercive, noncoercive, parabolic, Stokes and eigenvalue. In this work, that was originally motivated by problems involving classical periodic Schrödinger operators, an **error** **estimator** for the spectrum of positive self-adjoint operators is proposed, based on the problem residual and the definition of shifts. In particular, the aim is to estimate the **error** done when the problem finely discretized is projected on **a** low dimensional basis, giving rise to **a** coarsely discretized problem.

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HAL Id: cea-02509248
https://hal-cea.archives-ouvertes.fr/cea-02509248 Submitted on 16 Mar 2020
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The purpose of this Note is twofold. First, we propose **a** method to post-process the planewave approx- imation of the eigenmodes of (linear) periodic Schr¨ odinger operators and obtain **a** much better accuracy for **a** limited extra computational cost. This post-processing, although based on **a** simple application of the perturbation method, seems to be new. Note that this approach can be extended to the case of the (nonlinear) Gross-Pitaevski equation [3]. Second, we use this post-processing to construct an accurate **a** **posteriori** **error** **estimator** for the Gross–Pitaevskii equation. This **estimator** allows us to design an adaptive algorithm for solving this equation, which automatically refines the discretization along the convergence of the SCF iterations, by means of adaptive stopping criteria. The first steps of the SCF procedure are carried out in **a** coarse discretization space (it is indeed **a** loss of time to use **a** fine discretization space far from the minimizer), and the cut-off N is gradually increased to balance the two components of the total **error**: the discretization **error** on the one hand, and the **error** due to the fact that the SCF iterations has not converged yet on the other hand. The extension of this approach to more complex nonlinear Schr¨ odinger equations and to the Kohn–Sham model is work in progress.

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the size of the largest element in the mesh, so that **a** large part of the mesh can be refined before entering the resolved regime.
7. Conclusions
We have proposed **a** novel **a** **posteriori** **error** **estimator** for the Helmholtz problem with mixed boundary conditions in two and three space dimensions. The **estimator** is based on equilibrated flux reconstruction that relies on the solution of patchwise mixed finite element problems. It is reliable, where the reliability constant depends on the approximation factors σ ba and (possibly) σ e ba and tends to one when σ ba , e σ ba → 0, so that the **estimator** becomes asymptotically unknown-constant-free. We have also proven, via arguments based on elliptic regularity shift, that the conditions σ ba , e σ ba → 0 are met, for most situations of practical interest, when h p → 0 with fixed k. Finally, we have proven that the derived **estimator** is locally efficient and polynomial-degree-robust in all regimes and wavenumber-robust in the asymptotic regime σ ba ≤ 1.

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Keywords. 3D metal forming processes, Automatic adaptive remeshing, **A**-**posteriori** **error** **estimator**, Transfer
techniques, Equilibrated process
1 INTRODUCTION
For **a** large class of problems such as metal forming process like forging, rolling, cutting, etc., the use of the finite element method is still **a** challenging problem. It can typically involve high strain localization, large inelastic deformations and high temperature problems. Often in these types of application, mesh becomes unacceptable due to severe distortion or complex workpiece-die contact occurring which induces errors in the key internal variables (plastic strains and stresses) [Hamel 00] . As consequence, the ability to achieve **a** proper analysis with reasonable CPU cost is entirely dependent on the FE mesh spacing. Indeed, optimal mesh configuration changes continuously throughout the metal forming process. Consequently, the final results obtained with the use of **a** fixed mesh are unreliable and may give **a** rise to severe numerical difficulties and may even make it impossible to pursue the calculations any further because of severe element distortion. In order to overcome these difficulties and continue the simulation for large deformation cases, successive mesh adaptation is needed during the numerical simulation in order to obtain an optimal discretization according to the geometrical shape or/and physical solution. The main ingredients in any adaptive procedure are **a** **posteriori** **error** estimation and **a** mesh generator [Diez 07] :

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with order h s has been observed in [12]; this motivates the study of **a** **posteriori** **error** estimators that could efficiently drive an adaptive refinement strategy.
For the system (1.1)–(1.3), **a** **posteriori** **error** estimations for conforming Lagrange finite element methods (FEM) are now very common. The reader is referred to, e.g., [2, 5, 27] in which several types of estimators are detailed. In the residual based estimators, the main terms are inter-element jumps of the normal components of the gradients of the computed solution, weighted by constants whose explicit computa- tion was performed in [7] and [28]. Efficiencies of the estimators obtained in [7] vary, according to the problems, between 30 and 70, and between 1.5 and 3.5 if one numer- ically evaluates eigenvalues of some vertex centered local problems, as reported in [8]. References for non-conforming FEM may be found in [3] and for mixed FEM in [29]. The case of cell-centered FVM has been less studied, on the one hand because of their more recent use for elliptic problems, and, on the other hand, because they generally lack **a** discrete variational formulation. For the basic ”four point” scheme on so-called ”admissible” triangular meshes (see [14, 16]), Agouzal and Oudin [1] have used the connection of this scheme with mixed finite elements to derive an **a** **posteriori** **estimator** for the L 2 norm of the **error**; this **estimator** is not an upper bound for the **error**, but is asymptotically exact under mild hypothesis. **A** second **estimator** for this scheme has been given by Nicaise in [20]. This **estimator** is shown to be equivalent to the (broken) energy norm of the difference between the exact solution and an el- ementwise second order polynomial (globally discontinuous) reconstructed numerical solution. Then, in [21], Nicaise extends his ideas to the so-called ”diamond-cell” FVM (as described in [10]) and proposes an **a** **posteriori** **error** **estimator** which may be used if the cells of the mesh are triangles or rectangles (or tetrahedrons in dimension three). This **estimator** is completely computable (no unknown constant) and its efficiency is around 7 for the tests performed in [21]. Finally, Nicaise has extended his work to diffusion-convection-reaction equations in [22]. More recently, Vohral´ık [31] has also proposed **a** fully computable **a** **posteriori** **error** **estimator** for numerical approximations of diffusion-convection-reaction equations by cell-centered FVM on general meshes. The main improvement over [21, 22] is the asymptotic exactness of the **error** bound which, like in [21], measures the energy norm of the difference between the exact solution and **a** reconstructed, globally discontinuous, elementwise second order poly- nomial numerical solution. Note that in [22] the reconstructed numerical solution is globally continuous and may involve higher order polynomials on each element.

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In this paper, we are interested in **a** guaranteed and explicitly computable **a** **posteriori** **error** **estimator** based on an equilibration technique for the harmonic magnetodynamic formulations of the Maxwell system. Basically the equilibration is realized through the non-verification property of the constitutive laws equations. Some residual **a** **posteriori** **error** estimators have already been established for the **A** − ϕ magnetoharmonic formulation [18] as well as for the T − Ω one [19]. Now, we aim to derive an equilibrated **a** **posteriori** **estimator** for both of these dual problems, in the same philosophy of [27], where an equilibrated **estimator** is proposed in the magnetostatic framework. In other words, the **estimator** will be built starting from both formulations and will be available to estimate the sum of the errors of both resolutions. The originality of this work resides in the fact that, neglecting some higher order terms, the equivalence between the **error** and the **estimator** is proved, without the interference of unknown constants. So, we not only derive **a** reliable upper bound for the **error**, but also **a** lower one, where the constant is one up to higher order terms. Moreover, **a** local in space efficiency without unknown constants is proved, this result being useful for developing some adaptive mesh refinement algorithms.

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Several works deal with **a** **posteriori** **error** estimation for Galerkin discretizations of (1.2), in- cluding reliability and efficiency proofs [4, 32, 37], as well as convergence analysis of adaptive strategies [45]. However, to the best of our knowledge, these results are not “frequency-explicit”, in the sense that the reliability and efficiency constants implicitly depend on the frequency. As shown for instance in [11], the reliability constant may be surprisingly large in some cases, leading to important underestimation of the **error** (we also refer the reader to the numerical experiments presented in Section 5). This is especially problematic when the **estimator** is used as **a** stopping criterion. On the other hand, the convergence speed of adaptive schemes also depends on this constants. As **a** result, it is of interest to estimate the size of these constants, and to identify mesh sizes for which the **estimator** can be trusted.

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Section 4 is then devoted to our **a** **posteriori** **error** estimates. They consist of several independent estimators, the principal of which penalize the fact that ˜ p h is nonconforming and that its residual is nonzero. For pure diffusion problems, only these estimators (plus possibly still **a** Neumann boundary one) are present. When there is some convection, additional convection and upwinding estimators appear, and for cases with reaction, **a** reaction quadrature **estimator** may be present as well. We next prove that the principal (nonconformity, convection, and residual) estimators represent local lower bounds for the **error** as well, where in particular the efficiency constants are of the form c1 + c 2 min Pe , ρ , where Pe (the local P´eclet number) and ρ are given below by ( 4.12 ) and where c1 , c 2 only depend on local variations in S (i.e., on local inhomogeneities and anisotropies), on local vari- ations in w and r, on the space dimension, on the polynomial degree of ˜ p h , f , w, r, and on the shape-regularity parameter of the mesh. These estimators are thus in particular optimally efficient as the local P´eclet number gets sufficiently small. We are not able to obtain similar results for the upwinding **estimator** but numerical ex- periments suggest that this **estimator** represents **a** higher-order term as soon as the local P´eclet number gets sufficiently small. **A** more detailed discussion, as well as several other remarks, is given in Section 4.3 .

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The present parabolic potential and flux reconstructions are inspired from those derived in the context of **a** **posteriori** **error** estimates for elliptic problems in [12, 13, 30, 31, 32, 33]. The idea to derive parabolic **a** **posteriori** **error** estimates from elliptic estimates on each time level is rather natural. In fact, the residual-based **a** **posteriori** **error** estimates for conforming finite elements derived in [29, 7] take this form. We also mention [21, 20, 11] for the so-called elliptic reconstruction technique allowing for optimal **error** estimates in higher order norms for conforming finite elements. An important conceptual difference is that we reconstruct the (vector-valued) flux and that this quantity is discrete, is constructed locally by postprocessing, and is directly used to evaluate the **estimator**. In [6, 5, 27], various estimators for elliptic problems are extended to the heat equation in **a** conforming setting to bound the **error** mea- sured in the L 2 -norm in the space–time cylinder plus **a** time-weighted energy-norm;

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The purpose of this paper is to derive **a** **posteriori** **error** estimates, for such **a** model in order to introduce an auto- adaptive mesh refinement technique.
Since the pioneering work of Babushka and Reinbold [3], much has been written about adaptive methods for finite el- ement approximation with emphasis on both theoretical and computational aspects of the method. Several **a** **posteriori** **error** estimators for mixed finite element discretization of elliptic problems have been derived. For the residual-based estimators, which is the type of **estimator** that we use here, we can distinguish two types of estimation: the first of these was introduced by Braess and Verf¨urth [9] and gives bounds on the **error** in **a** mesh dependent norm which is close to the energy norm of the continuous problem in its primal form. In the presense of **a** saturation assumption (which is not always satisfied) this **estimator** is reliable and efficient in this norm, but somehow it is not efficient in the natural norm of the mixed formulation. This estimate was improved by Lovadina and Stenberg [20] and Larson and Malqvist [18] by introducing **a** postprocessing technique.

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also pay **a** spe
ial attention to the
ase of **a** heterogeneous and anisotropi
diusion tensor
K ; it turns out that some fully robust results with respe
t to diusion heterogenei ties
an
be obtained for our new diusive ux **estimator** for **a**
ertain
lass of DG s
hemes su
h
as those introdu
ed by Ern, Stephansen and Zunino [51℄. These s
hemes use diusivity-

136 En savoir plus

As **a** starting point, we consider the classical diffusion equation and observe that, although the approach presented in [18, Chapter 8] is guaranteed, it remains difficult to prove the local efficiency of the **estimator**. We address this issue by proposing **a** **posteriori** estimators that are guaranteed and locally efficient.
We focus on Cartesian meshes since such structures are relevant in nuclear core applications, and outline **a** robust marker strategy for this specific constraint, the direction marker strategy. We observe numerically that the AMR strategy is sensitive to the choice of the threshold parameter. We compare various **a** **posteriori** estimators under different criteria. We show that the choice of the reconstruction has **a** strong influence on the AMR strategy. The post-processing approaches are shown to be more efficient than the average reconstruction. In the case of the lowest-order Raviart-Thomas-Nédélec finite element, the RTN 0 post-processing gives **a**

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Several works in the literature have highlighted the great potential of coupling **a** **posteriori** **error** estimators to shape optimization algorithms. In the pioneering work [7], the authors identify two diﬀerent sources for the numerical **error**: on the one hand, the **error** arising from the approximation of the diﬀerential problem and on the other hand, the **error** due to the approximation of the geometry. Starting from this observation, Banichuk et al. present **a** ﬁrst attempt to use the information on the discretization of the diﬀerential problem provided by **a** recovery-based **estimator** and the **error** arising from the approximation of the geometry to develop an adaptive shape optimization strategy. This work has been later extended by Morin et al. in [28], where the adaptive discretization of the governing equations by means of the Adaptive Finite Element Method is linked to an adaptive strategy for the approximation of the geometry. The authors derive estimators of the numerical **error** that are later used to drive an Adaptive Sequential Quadratic Programming algorithm to appropriately reﬁne and coarsen the computational mesh. Several other authors have used adaptive techniques for the approximation of PDE’s in order to improve the accuracy of the solution and obtain better ﬁnal conﬁgurations in optimal structural design problems. We refer to [3, 25, 34, 36] for some examples. We remark that in all these works, **a** **posteriori** estimators only provide qualitative information about the numerical **error** due to the discretization of the problems and are essentially used to drive mesh adaptation procedures. To the best of our knowledge, no guaranteed fully-computable estimate has been investigated and the **error** in the shape gradient itself is not accounted for, thus preventing reliable stopping criteria to be derived.

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new element density. Among the very numerous proposals that were, and still are available, three groups can be classified, depending on whether they are based on: the estimators based on residuals analysis, introduced by Babuska and Rheiboldt [ 14 ], the estimators based on the concept of **error** in constitu- tive relation, initiated by Ladvéze et al. [ 15 ], and the estima- tors based on smoothing techniques developed by Zienkiewicz et Zhu [ 16 ].Although the approach of the residual **estimator** proposed by Babuška and Rheinbold [ 14 ] is mathe- matically rigorous, its extension to 3D nonlinear problems is facing some major challenges. Indeed the efficiency of this **estimator** depends on the mesh quality and regularity of the solution, which makes it less suitable for large deformation problems. In the literature, the majority of applications are presented in the context of 1D and 2D academic problems. The key point of the proposed **estimator** by Ladevèze et al. [ 15 ] is the construction of admissible displacement-stress pair. For the sake of simplicity, in the context of FE method with compressible material, the displacement field can be consid- ered as admissible. Conversely, the stress field is not statically admissible. In the literature, Sever method can be used to calculate this admissible stress [ 17 ]. In spite of its remarkable efficiency, this technique has been rarely used in FE simula- tion software because of the cost of its implementation. In this work, the **estimator** based on smoothing techniques is used due to its cost effectiveness and reliability to estimate errors and for its simplicity of implementation. The main idea of the smoothing techniques is based on the construction of **a** recov- ered stress tensor field ~σ more accurate than the finite element solution σ h . This recovered stress tensor field ~σ,can be con-

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The residual-based **error** **estimator** has been developed for an approxi- mation in the stochastic dimension based on **a** truncated polynomial chaos expansion. The strategy can also be applied for other kinds of approxima- tion spaces based on wavelets or piecewise polynomials for example. The proposed **error** **estimator** can be used to compare the accuracy of the dif- ferent approximation spaces and numerical methods by evaluating the two parts of the **estimator** (spatial and stochastic one). It enables also to make **a** relative comparison in terms of accuracy between the spatial errors and the stochastic errors.

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Figure 3: Evolution of the mean value of the **estimator** and of the MCSM estimated stochastic **error**.
According to Fig. 3, we can deduce that : -The **estimator** and the stochastic **error** estimated by the MSCM are very close. -While the accuracy level of the solution of the SSFEM matrix system is low (ε is greater than 10 −4 in Fig. 3) the stochastic **error** is the same with di fferent orders of PCE. - While the accuracy level is quite high (ε is lower than 10 −4 in Fig. 3) **a** higher order of PCE yields **a** smaller stochastic **error**. -With **a** given order of PCE, when the accuracy level increases, the evolution of the stochastic **error** decreases up to **a** given value before being stable (log(ε) = −6 with order p = 2 and log(ε) = −8 with order p = 4). It is thus wasteful to increase the accuracy level of the solution of the SSFEM matrix system beyond these points.

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HAL Id: hal-02321140
https://hal.inria.fr/hal-02321140
Submitted on 21 Oct 2019
HAL is **a** multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

The discrete problem amounts to **a** system of nonlinear equations, and, in practice, is solved using an iterative method involving some kind of linearization. Given an approximate solution, say u L,h , at **a** given stage of the iterative process and on **a** given mesh, there are actually two sources of **error**, namely linearization and discretization. Balancing these two sources of **error** can be of paramount importance in practice, since it can avoid performing an excessive number of nonlinear solver iterations if the discretiza- tion **error** dominates. Therefore, the second objective of this work is to design **a** **posteriori** **error** estimates distinguishing linearization and discretization errors in the context of an adaptive procedure. This type of analysis has been started by Chaillou and Suri [11, 12] for **a** certain class of nonlinear problems similar to the present one and in the context of iterative solution of linear algebraic systems in [21]. Chaillou and Suri only considered **a** fixed stage of the linearization process, while we take here the analysis one step further in the context of an iterative loop. Furthermore, they only considered **a** specific form for the linearization, namely of quasi-Newton type, while we allow for **a** wider choice, including Newton– Raphson methods. We consider an adaptive loop in which at each step, **a** fixed mesh is considered and the nonlinear solver is iterated until the linearization **error** estimate is brought below the discretization **error** estimate; then, the mesh is adaptively refined and the loop is advanced. In this work, we will not tackle the delicate issue of proving the convergence of the above adaptive algorithm. We will also assume that at each iterate of the nonlinear solver, **a** well-posed problem is obtained. This property is by no means granted in general; it amounts, for the p-Laplacian, to assume, as mentioned before in [9], that the gradient norm of the approximate solution is positive everywhere in the domain. We mention that in our numerical experiments, all the discrete problems were indeed found to be well-posed.

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Engineering, Queen’s Building, The Parade, Cardiff CF24 3AA Wales, UK.
3 EADS Corporate International Chair. ´ Ecole Centrale de Nantes, Nantes, France.
Abstract
In this paper **a** new technique aimed to obtain accurate estimates of the **error** in energy norm using **a** moving least squares (MLS) recovery-based pro- cedure is presented. We explore the capabilities of **a** recovery technique based on an enhanced MLS fitting, which directly provides continuous interpolated fields, to obtain estimates of the **error** in energy norm as an alternative to the superconvergent patch recovery (SPR). Boundary equilibrium is enforced using **a** nearest point approach that modifies the MLS functional. Lagrange multipliers are used to impose **a** nearly exact satisfaction of the internal equi- librium equation. The numerical results show the high accuracy of the pro- posed **error** **estimator**.

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