A posteriori error estimation for nonlinear variational problems by duality theory

Repin, Sergey

doi:10.1007/bf02673600

Cited by 71 publications

(83 citation statements)

References 14 publications

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“…The second part of the right-hand side of (36) is formed by the functional L τ,f whose value is estimated from above by the quantity | | | L τ,f | | | ε(w) . From (36), (36), and (37) it follows that…”

Section: Upper Bound Of the Error Formentioning

confidence: 98%

“…For elliptic type problems of the divergent type functional a posteriori estimates were derived in [36,37,38,39,40,43,44] and some other papers with the help of duality methods in the calculus of variations (see [30] for a consequent exposition of the approach). Computable upper bounds of approximation errors for the Stokes problem with Dirichlét boundary conditions were derived by this method in [41] and for some classes of generalized Newtonian fluids in [21,40].…”

Section: Introductionmentioning

confidence: 99%

“…Estimate (1) is one of the simplest a posteriori estimates of the functional type (for the equation divA∇u + f = 0 with positive definite symmetric matrix A such estimates are presented in [36,37]). It is easy to observe that the right-hand side of (1) is nonnegative and vanishes if and only if v = u and τ = ∇u.…”

Section: Introductionmentioning

confidence: 99%

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Two-sided a posteriori estimates for a generalized Stokes problem

Repin

Stenberg²

2009

J Math Sci

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The paper is concerned with deriving computable majorants and minorants of the difference between the exact solution of for the so-called three-field formulation of the generalized Stokes problem and any functions from the admissible (energy) spaces that contain velocity, pressure and stress fields. Physical motivation of this problem is related to models of viscous fluids with polymeric chains. For the the case of uniform Dirichlét boundary conditions this model and respective numerical approximation methods were analyzed in [14]. In the present paper, we consider the generalized Stokes problem with mixed Dirichlét/Neumann boundary conditions and variable viscosity in the context of a posteriori error analysis. For the velocity, pressure, and stress fields we derive two-sided functional a posteriori error estimates. The estimates are practically computable, sharp (i.e., have no gap between the left-and right-hand sides), and are valid for arbitrary functions from the respective functional classes. The estimates are derived by transformations of the integral identity that defines the solution (this method was suggested and used in [39,40] for certain classes of elliptic type problems). Error majorants are given by weighted sums of the terms that present penalties for violations of all the relations of the problem considered with the weights defined by the constants in the Friederichs-Poincáre and Ladyzhenskaja-BabuskaBrezzi inequalities, respectively.

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Section: Upper Bound Of the Error Formentioning

confidence: 98%

Section: Introductionmentioning

confidence: 99%

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Two-sided a posteriori estimates for a generalized Stokes problem

Repin

Stenberg²

2009

J Math Sci

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“…Letỹ be any function from the admissible set Y := H 1 0 (Ω), which we view as an approximation of the solution of the elliptic problem (2a)-(2b). It was shown (see, e.g., [11] and [12]) that the error of the approximationỹ satisfies the following estimate:…”

Section: Introductionmentioning

confidence: 99%

A Posteriori Estimates for Cost Functionals of Optimal Control Problems

Gaevskaya

Hoppe

Repin

Numerical Mathematics and Advanced Applications

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“…Our method differs from these approaches and its derivation is based on our previous publications (see [19][20][21][22][23][24][25][26][27][28][29][30][31]), in which estimates of the difference between the exact solution of boundary value problems and arbitrary functions from the corresponding energy space has been derived by purely functional methods without requiring specific information on the approximating subspace and the numerical method used. As a result, the estimates contain no mesh dependent constants and are valid for any conforming approximation from the respective energy space.…”

Section: Introductionmentioning

confidence: 99%