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61
Adjoint Recovery of Superconvergent Functionals from Approximate Solutions of Partial Differential Equations
, 1998
"... Abstract. Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that ..."
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Cited by 86 (12 self)
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Abstract. Motivated by applications in computational fluid dynamics, a method is presented for obtaining estimates of integral functionals, such as lift or drag, that have twice the order of accuracy of the computed flow solution on which they are based. This is achieved through error analysis that uses an adjoint PDE to relate the local errors in approximating the flow solution to the corresponding global errors in the functional of interest. Numerical evaluation of the local residual error together with an approximate solution to the adjoint equations may thus be combined to produce a correction for the computed functional value that yields the desired improvement in accuracy. Numerical results are presented for the Poisson equation in one and two dimensions and for the nonlinear quasionedimensional Euler equations. The theory is equally applicable to nonlinear equations in complex multidimensional domains and holds great promise for use in a range of engineering disciplines in which a few integral quantities are a key output of numerical approximations. Key words. PDEs, adjoint equations, error analysis, superconvergence AMS subject classifications. 65G99, 76N15 PII. S0036144598349423
Bounds for LinearFunctional Outputs of Coercive Partial Differential Equations: Local Indicators and Adaptive Refinement
, 1997
"... this paper we focus on three new developments. First, we introduce a modified energy objective, and hence modified Lagrangian, that permits both more transparent interpretation and more ready generalization. Second, we demonstrate that the bound gap  the difference between the upper and lower bou ..."
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Cited by 44 (9 self)
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this paper we focus on three new developments. First, we introduce a modified energy objective, and hence modified Lagrangian, that permits both more transparent interpretation and more ready generalization. Second, we demonstrate that the bound gap  the difference between the upper and lower bounds for the desired output  can be represented as the sum of positive contributions  local indicators  associated with the elements TH of TH . Third, based on these local boundgap error indicators, we develop adaptive strategies by which to reduce the bound gap  and hence improve our validated prediction for the output of interest  through optimal refinement of TH . The resulting method is applied to an illustrative problem in linear elasticity.
A posteriori error estimation techniques in practical finite element analysis
, 2005
"... In this paper we review the basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem. We give the basic ideas to establish global error estimates for the energy norm as well as goaloriented error estimates. While we show how these er ..."
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Cited by 41 (5 self)
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In this paper we review the basic concepts to obtain a posteriori error estimates for the finite element solution of an elliptic linear model problem. We give the basic ideas to establish global error estimates for the energy norm as well as goaloriented error estimates. While we show how these error estimation techniques are employed for our simple model problem, the emphasis of the paper is on assessing whether these procedures are ready for use in practical linear finite element analysis. We conclude that the actually practical error estimation techniques do not provide mathematically proven bounds on the error and need to be used with care. The more accurate estimation procedures also do not provide proven bounds that, in general, can be computed efficiently. We also briefly comment upon the state of error estimations
Analytic adjoint solutions for the quasionedimensional Euler equations
 J. Fluid Mechanics
, 2001
"... The analytic properties of adjoint solutions are examined for the quasionedimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is requir ..."
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Cited by 34 (9 self)
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The analytic properties of adjoint solutions are examined for the quasionedimensional Euler equations. For shocked flow, the derivation of the adjoint problem reveals that the adjoint variables are continuous with zero gradient at the shock, and that an internal adjoint boundary condition is required at the shock. A Green’s function approach is used to derive the analytic adjoint solutions corresponding to supersonic, subsonic, isentropic and shocked transonic flows in a converging–diverging duct of arbitrary shape. This analysis reveals a logarithmic singularity at the sonic throat and confirms the expected properties at the shock. 1.
Improved lift and drag estimates using adjoint Euler equations
 AIAA Paper
, 1999
"... This paper demonstrates the use of adjoint error analysis to improve the order of accuracy of integral functionals obtained from CFD calculations. Using second order accurate finite element solutions of the Poisson equation, fourth order accuracy is achieved for two different categories of functiona ..."
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Cited by 32 (8 self)
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This paper demonstrates the use of adjoint error analysis to improve the order of accuracy of integral functionals obtained from CFD calculations. Using second order accurate finite element solutions of the Poisson equation, fourth order accuracy is achieved for two different categories of functional in the presence of both curved boundaries and singularities. Similarly, numerical results for the Euler equations obtained using standard second order accurate approximations demonstrate fourth order accuracy for the integrated pressure in two quasi1D test cases, and a significant improvement in accuracy in a twodimensional case. This additional accuracy is achieved at the cost of an adjoint calculation similar to those performed for design optimization. 1 Introduction In aeronautical CFD, engineers desire very accurate prediction of the lift and drag on aircraft, but they are less concerned with the precise details of the flow field in general, although there is a clear need to underst...
A posteriori error estimates for higher order godunov finite volume methods on unstructured meshes
 Complex Applications III, R. Herbin and D. Kroner (Eds), HERMES Science Publishing Ltd
, 2002
"... A posteriori error estimates for high order Godunov finite volume methods are presented which exploit the two solution representations inherent in the method, viz. as piecewise constants u0 and cellwise pth order reconstructed functions R 0 pu0. Using standard duality arguments, we construct exact ..."
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Cited by 28 (1 self)
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A posteriori error estimates for high order Godunov finite volume methods are presented which exploit the two solution representations inherent in the method, viz. as piecewise constants u0 and cellwise pth order reconstructed functions R 0 pu0. Using standard duality arguments, we construct exact error representation formulas for derived functionals that are tailored to the class of high order Godunov finite volume methods with data reconstruction, R 0 pu0. We then devise computable error estimates that exploit the structure of Godunov finite volume methods. The present theory applies directly to a wide range of finite volume methods in current use including MUSCL, TVD, UNO, and ENO methods. Issues such as the treatment of nonlinearity and postprocessing of dual (adjoint) problem data are discussed. Numerical results for linear advection and nonlinear scalar conservation laws at steadystate are presented to validate the analysis.
A Posteriori Error Analysis And Adaptivity For Finite Element Approximations Of Hyperbolic Problems
, 1997
"... this article is to present an overview of recent developments in the area of a posteriori error estimation for finite element approximations of hyperbolic problems. The approach pursued here rests on the systematic use of hyperbolic duality arguments. We also discuss the question of computational ..."
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Cited by 25 (4 self)
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this article is to present an overview of recent developments in the area of a posteriori error estimation for finite element approximations of hyperbolic problems. The approach pursued here rests on the systematic use of hyperbolic duality arguments. We also discuss the question of computational implementation of the a posteriori error bounds into adaptive finite element algorithms
On adjoint equations for error analysis and optimal grid adaptation in CFD
, 1997
"... This paper explains how the solutions of appropriate adjoint equations can be used to estimate the errors in important integral quantities, such as lift and drag, obtained from CFD computations. These error estimates can be used to obtain improved estimates of the integral quantities, or as the basi ..."
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Cited by 21 (3 self)
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This paper explains how the solutions of appropriate adjoint equations can be used to estimate the errors in important integral quantities, such as lift and drag, obtained from CFD computations. These error estimates can be used to obtain improved estimates of the integral quantities, or as the basis for optimal grid adaptation. The theory is presented for both finite volume and finite element approximations. For a nodebased finite volume discretisation of the Euler equations on unstructured grids, the adjoint analysis makes it possible to prove second order accuracy. A superconvergence property is proved for a finite element discretisation of the Laplace equation, and references are provided for the extension of the analysis to the convection/diffusion and incompressible NavierStokes equations. This paper was presented at the symposium Computing the Future II: Advances and Prospects in Computational Aerodynamics to honour the contributions of Prof. Earll Murman to CFD and the aerospace community. Key words and phrases: error analysis, adjoint equations, grid adaptation
Asymptotic a Posteriori Finite Element Bounds for the Outputs of Noncoercive Problems: the Helmholtz and Burgers Equations
, 1998
"... We describe an a posteriori finite element procedure for the efficient computation of lower and upper estimators for linearfunctional outputs of noncoercive linear and semilinear elliptic secondorder partial differential equations. Under a relatively weak hypothesis related to the relative magni ..."
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Cited by 16 (4 self)
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We describe an a posteriori finite element procedure for the efficient computation of lower and upper estimators for linearfunctional outputs of noncoercive linear and semilinear elliptic secondorder partial differential equations. Under a relatively weak hypothesis related to the relative magnitude of the L 2 and H 1 errors of the reconstructed solution, these lower and upper estimators converge to the true output from below and above, respectively, and thus constitute asymptotic bounds. In numerical experiments we find that our hypothesis is satisfied once the finite element triangulation even roughly resolves the structure of the exact solution, and thus, in practice, the bounds prove quite reliable. Numerical results are presented for the onedimensional Helmholtz equation and for the Burgers equation. 1 Introduction In a recent series of papers [10, 11, 12, 13] we introduce an a posteriori finite element method for the efficient computation of strict upper and lower boun...