Error estimates for the approximation of semicoercive variational inequalities

Spann, W

doi:10.1007/s002110050082

Cited by 9 publications

(13 citation statements)

References 16 publications

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“…Although there is much work on the error analysis for finite element approximations to both steady and evolutionary obstacle problems [1,3,7,12,18,19,23,28,44,45,38,50,53,56,58,59,61,63], error estimates of the numerical methods for the Bingham fluid have not been investigated as much. Two well-known approaches to solve the Bingham fluid are the regularization method and the Augmented Lagrangian method (see section 6 for references).…”

Section: Introductionmentioning

confidence: 99%

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

Zhang

2003

Numerische Mathematik

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The flow of a Bingham fluid in a cylindrical pipe can give rise to free boundary problems. The fluid behaves like a viscous fluid if the shear stress, expressed as a linear function of the shear rate, exceeds a yield value, and like a rigid body otherwise. The surfaces dividing fluid and rigid zones are the free boundaries. Therefore the solution for such highly nonlinear problems can in general only be obtained by numerical methods. Considerable progress has been made in the development of numerical algorithms for Bingham fluids [2,20,27,29,30,32]. However, very little research can be found in the literature regarding the rate of convergence of the numerical solution to the true continuous solution, that is the error estimate of these numerical methods. Error estimates are a critically important issue because they tells us how to control the error by appropriately choosing the grid sizes and other related parameters. This paper concerns the error estimates of a unsteady Bingham fluid modeled as a variational inequality due to Duvaut-Lions [16] and Glowinski [30]. The difficulty both in the analytical and numerical treatment of the mathematical model is due to the fact that it contains a nondifferentiable term. A common technique, called the regularization method, is to replace the non-differentiable term by a perturbed differentiable term which depends on a small regularization parameter . The regularization method effectively reduces the variational inequality to an equation (a regularized problem) which is much easier to cope with. This paper has achieved the following. (1) Error estimates are derived for a continuous time Galerkin method in suitable norms. (2) We give an estimate of the difference between the true solution and the regularized solution in terms of . (3) Some regularity properties for both regularized solution and the true solution are proved. (4) The error 154 Y. Zhangestimates for full discretization of the regularized problem using piecewise linear finite elements in space, and backward differencing in time are established for the first time by coupling the regularization parameter and the discretization parameters h and t. (5) We are able to improve our estimates in the one-dimensional case or under stronger regularity assumptions on the true solution. The estimates for the one-dimensional case are optimal and confirmed by numerical experiment. The estimates from (4) and (5) provide very important information on the measure of the error and give us a powerful mechanism to properly choose the parameters h, t and in order to control the error (see Corollary 4.4). The above estimates extend the error bounds derived in Glowinski, Lions and Trémolières [32] (chapter 5, pp. 348-404) for the stationary Bingham fluid to the time-dependent one, which is the main contribution of this paper. (2000): 35k85, 65M15, 65M60, 76A05, 76M10 Mathematics Subject Classification

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Section: Introductionmentioning

confidence: 99%

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

Zhang

2003

Numerische Mathematik

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“…Generalization of this approach to semi-coercive problems of variational form has later been obtained by Gwinner [11], [12], [13]. See also Hlavacek [19], [20], Panagiotopoulos [27], Hlavacek, Haslinger, Necas and Lovisek [21], Kaplan and Tichatschke [22] and Spann [31] for related results. However, to the authors knowledges, a common convergence theory applicable to the numerical study of a large class of semi-coercive unilateral problems allowing rigid body motions does not exist in the literature.…”

Section: Introductionmentioning

confidence: 93%

A discretization theory for a class of semi-coercive unilateral problems

Adly

Goeleven

2000

Numerische Mathematik

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In this paper, we present a convergence analysis applicable to the approximation of a large class of semi-coercive variational inequalities. The approach we propose is based on a recession analysis of some regularized Galerkin schema. Finite-element approximations of semi-coercive unilateral problems in mechanics are discussed. In particular, a SignoriniFichera unilateral contact model and some obstacle problem with frictions are studied. The theoretical conditions proved are in good agreement with the numerical ones. Subject Classification (1991): 49J40, 49J52, 35J20 Mathematics

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“…Note that they are based on the standard techniques used to obtain error estimates for the approximation of variational inequalities (see [10,20,21]). …”

Section: Error Estimates For the Fully Discretized Schemementioning

confidence: 99%

“…Here, our interest is to obtain an abstract error estimate for the approximate problem. To this aim, we will focus on the variational inequality, which defines the optimal control, and its discrete counterpart so that the results derived by Falk [10] and Spann [20,21] can be extended to our problem.…”

Section: Introductionmentioning

confidence: 99%

Error Estimates for the Approximation of a Class of Optimal Control Systems Governed by Linear PDEs

Gamallo

Hernández

2009

Numerical Functional Analysis and Optimization

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This paper deals with a class of optimal control problems in which the system is governed by a linear partial differential equation and the control is distributed and with constraints. The problem is posed in the framework of the theory of optimal control of systems. A numerical method is proposed to approximate the optimal control. In this method, the state space as well as the convex set of admissible controls are discretized. An abstract error estimate for the optimal control problem is obtained that depends on both the approximation of the state equation and the space of controls. This theoretical result is illustrated by some numerical examples from the literature.

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Error estimates for the approximation of semicoercive variational inequalities

Cited by 9 publications

References 16 publications

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method

A discretization theory for a class of semi-coercive unilateral problems

Error Estimates for the Approximation of a Class of Optimal Control Systems Governed by Linear PDEs

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