A posteriori error estimates for nonlinear problems:Lr, (0,T;W1,ρ (Ω))-error estimates for finite element discretizations of parabolic equations

Verfürth, Rüdiger

doi:10.1002/(sici)1098-2426(199807)14:4<487::aid-num4>3.0.co;2-g

Cited by 45 publications

(36 citation statements)

References 16 publications

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“…Next, we extend the results to a semilinear heat equation where the diffusion coefficient now depends on the solution, as already considered in [EJ2], [NSV] and [Ve2], [Ve3], [Ve5]. Some arguments for the extension of a posteriori analysis to nonlinear problems have been proposed in [PR] and [Ve5]; however they do not seem appropriate for the present problem and for the kinds of indicators we work with.…”

Section: Introductionmentioning

confidence: 92%

“…However, it seems that the analogous results concerning parabolic problems are presently not complete. They most often deal either only with time scheme adaptivity (see [JNT] or [NSV]) or space finite element adaptivity (see for instance [BB1], [BB2], [BBHM] or [BM]) or with space-time finite element adaptivity (see [EJ1], [EJ2], [Ve2], [Ve3] and [Ve5]): the finite element discretization in these references relies on the space-time variational formulation of the equation and this leads to a family of error indicators which represent the combined space and time errors.…”

Section: Introductionmentioning

confidence: 99%

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A posteriori analysis of the finite element discretization of some parabolic equations

2004

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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 semi-linéaires, par un schéma d'Euler implicite en temps et paŕ eléments finis en espace. L'idée de cet article est de construire des indicateurs d'erreur liésà l'approximation en temps et en espace et de prouver leuŕ equivalence avec l'erreur, dans le but de travailler avec des pas de temps adaptatifs et des maillages d'éléments finis adaptésà la solution.

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Section: Introductionmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

A posteriori analysis of the finite element discretization of some parabolic equations

2004

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“…Various other a posteriori estimates for semidiscrete and fully discrete approximations to linear and nonlinear parabolic problems in various norms are found in the literature [1,4,5,11,16,25,26,31,32]. In particular, Babuška, Feistauer, anď Solín [4] have derived estimates in L 2 (0, T ; L 2 (Ω)); see also Babuška et al [1,5].…”

Section: Introductionmentioning

confidence: 99%

Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems

Makridakis¹,

Nochetto²

2003

SIAM J. Numer. Anal.

150

177

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Abstract. We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of L ∞ (0, T ; L 2 (Ω)) and the higher order spaces, L ∞ (0, T ; H 1 (Ω)) and H 1 (0, T ; L 2 (Ω)), with optimal orders of convergence.

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“…Rigorous a posteriori error estimates for nonlinear parabolic problems seem much less developed. In nondegenerate cases, Verfürth [46,47] was able to obtain an estimator which is both reliable and efficient. A pioneering contribution for degenerate parabolic problems has been obtained by Nochetto et al in [36].…”

Section: Introductionmentioning

confidence: 99%

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

2014

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We consider in this paper the time-dependent two-phase Stefan problem and derive a posteriori error estimates and adaptive strategies for its conforming spatial and backward Euler temporal discretizations. Regularization of the enthalpy-temperature function and iterative linearization of the arising systems of nonlinear algebraic equations are considered. Our estimators yield a guaranteed and fully computable upper bound on the dual norm of the residual, as well as on the L 2 (L 2 ) error of the temperature and the L 2 (H −1 ) error of the enthalpy. Moreover, they allow to distinguish the space, time, regularization, and linearization error components. An adaptive algorithm is proposed, which ensures computational savings through the online choice of a sufficient regularization parameter, a stopping criterion for the linearization iterations, local space mesh refinement, time step adjustment, and equilibration of the spatial and temporal errors. We also prove the efficiency of our estimate. Our analysis is quite general and is not focused on a specific choice of the space discretization and of the linearization. As an example, we apply it to the vertex-centered finite volume (finite element with mass lumping and quadrature) and Newton methods. Numerical results illustrate the effectiveness of our estimates and the performance of the adaptive algorithm.MSC: 65N08, 65N15, 65N50, 80A22 IntroductionThe two-phase Stefan problem models a phase change process which is governed by the Fourier law, c.f. Friedman [22]. The two phases, typically solid and liquid, are separated by a moving interface, whose motion is governed by the so-called Stefan condition., be an open bounded polygonal or polyhedral domain, not necessarily convex, and let T > 0. The mathematical statement of the problem is as follows: given an initial enthalpy u 0 and a source function f , find the enthalpy u such that(1.1c) * This work was supported by the ERT project "Enhanced oil recovery and geological sequestration of CO 2 : mesh adaptivity, a posteriori error control, and other advanced techniques" (LJLL/IFPEN) † daniele.di-pietro@univ-montp2.fr ‡

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A posteriori error estimates for nonlinear problems:Lr, (0,T;W1,ρ (Ω))-error estimates for finite element discretizations of parabolic equations

Cited by 45 publications

References 16 publications

A posteriori analysis of the finite element discretization of some parabolic equations

A posteriori analysis of the finite element discretization of some parabolic equations

Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems

Adaptive regularization, linearization, and discretization and a posteriori error control for the two-phase Stefan problem

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