A posteriori error estimation for the semidiscrete finite element method of parabolic differential equations

Babuška, Ivo; Ohnimus, S.

doi:10.1016/s0045-7825(00)00340-6

Cited by 27 publications

(29 citation statements)

References 18 publications

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“…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]. Verfürth [31,32] showed estimates in L r (0, T ; L ρ (Ω)), with 1 < r, ρ < ∞ for certain fully discrete approximations of certain quasilinear parabolic equations.…”

Section: Introductionmentioning

confidence: 99%

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Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems

Makridakis¹,

Nochetto²

2003

SIAM J. Numer. Anal.

155

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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Section: Introductionmentioning

confidence: 99%

“…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.

155

177

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“…The wear values process during the particular time units for c → 0, 0 < δ < 1 is convergent to zero (g ∞ → 0). We have [16]:…”

Section: A New Variable Methods For Wear Cumulative Process Determinationmentioning

confidence: 99%

“…1-3 by the in experimental way obtained dimensional wear values W 1g , W 2g in µm 3 respectively. The cumulative wear value after finite N or infinite time units of operation has finally the form (16).…”

Section: Control Contribution For Wear Bearing Recurrence Processmentioning

confidence: 99%

Control contribution for wear bearing recurrence process

Wierzcholski¹,

Miszczak²

2014

Bulletin of the Polish Academy of Sciences Technical Sciences

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Abstract. This paper presents the methods of control problem solutions using recurrence equations implementation and UOS transformation for the bearing wear estimation during the finite and infinite time units of an operation process. If we have two wear value increase processes then very important is information which process is divergent more slowly. On the other hand, in comparison with two convergent processes we must decide which process is convergent more quickly. The wear process is determined mostly by the summation factor method. Such a method is applied for the solutions of recurrence equations with variable coefficients.Key words: recurrences equations, wear bearing control process. PreliminariesThe problems referring to the experimental and numerical wear of slide bearing exploitation questions require the knowledge of information about the features of the sequence of the existing wear process during the operation time [1][2][3][4]. The abovementioned information includes especially the kind and velocity of the wear value increases during the particular time units of the operation and very important are convergence and divergence phenomena of the wear value increases. If we have two wear value divergent processes, then very important is information which process is divergent more slowly. On the other hand, in comparison with two convergent processes, we must decide which process is convergent more quickly. This paper allows us to consider such a problem. If the wear value process in particular time units of the operating time is decreasing to some wear limit values, then it is possible to obtain the wear bearing stabilization in sufficient many time units of the operation. If the wear value process in particular time units of the operating time is increasing, then we have the phenomenon of wear bearing continuous increments in sufficient many time units of the operation. In both cases mentioned above, the sum of wear values after the considered time units of the operation, i.e. the cumulative function of wear values can be divergent. The most important for practical needs is the convergence or divergence of the cumulative values of bearing wear after the considered time units of the operation. To perform investigations of the abovementioned control problems, the wear value processes are described by non-homogeneous recurrent equations with a variable free term. The results obtained in this paper show that convergence or divergence of the wear processes depends on experimentally determined coefficients which occur in the considered wear recurrent equations. The coefficients depend on bearing materials, operation conditions, environmental external influences, and vibration influences. Therefore, the assumption and knowledge of materials, exploitation conditions and finally external influences enable us to determine the control of wear during the operation time [3,4]. The results obtained in this paper show how to anticipate wear of a slide bearing after many time units of the operation if bearing mater...

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“…Most of these algorithms are based on a posteriori error estimators which provide appropriate tools for adaptive mesh refinements. The theory of a posteriori analysis of finite element methods for parabolic problems is well-developed (see, e.g., [3,4,19,22,27,29,36,40]). Surprisingly, there has been considerably less work on the error control of finite element methods for second order hyperbolic problems, despite the substantial amount of research in the design of finite element methods for the wave problem (see, e.g., [6,7,8,11,20]).…”

Section: Introductionmentioning

confidence: 99%

A POSTERIORI L^∞(L²)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL CONTROL PROBLEMS

Hou¹

2013

Bulletin of the Korean Mathematical Society

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Abstract. In this paper, we discuss the a posteriori error estimates of the semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by the order k Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order k(k ≥ 0). Using mixed elliptic reconstruction method, a posteriori L ∞ (L 2 )-error estimates for both the state and the control approximation are derived. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.

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A posteriori error estimation for the semidiscrete finite element method of parabolic differential equations

Cited by 27 publications

References 18 publications

Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems

Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems

Control contribution for wear bearing recurrence process

A POSTERIORI L^∞(L²)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL CONTROL PROBLEMS

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A posteriori error estimation for the semidiscrete finite element method of parabolic differential equations

Cited by 27 publications

References 18 publications

Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems

Elliptic Reconstruction and a Posteriori Error Estimates for Parabolic Problems

Control contribution for wear bearing recurrence process

A POSTERIORI L∞(L2)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL CONTROL PROBLEMS

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A POSTERIORI L^∞(L²)-ERROR ESTIMATES OF SEMIDISCRETE MIXED FINITE ELEMENT METHODS FOR HYPERBOLIC OPTIMAL CONTROL PROBLEMS