Richardson Extrapolation and Defect Correction of Finite Element Methods for Optimal Control Problems

Liu, Tang

doi:10.4208/jcm.2009.09-m1001

Cited by 5 publications

(3 citation statements)

References 18 publications

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“…The study for superconvergence of finite element methods is a topic of importance. See, for example, [1][2][3][4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Global Superconvergence Estimates of Petrov-Galerkin Methods for Hyperbolic Equations

Yang

2010

2010 2nd International Workshop on Database Technology and Applications

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In this paper, Petrov-Galerkin methods are employed for numerical solutions to hyperbolic partial differential equations, which are important mathematical models with many applications in engineering, such as wave transmission, petroleum reservoir. By virtue of superclose analysis and an interpolation postprocessing technique, the global superconvergence is obtained, which is new even for elliptic partial differential equations.

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“…The study for superconvergence of finite element methods is a topic of importance. See, for example, [1][2][3][4][5][6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

Global Superconvergence Estimates of Petrov-Galerkin Methods for Hyperbolic Equations

Yang

2010

2010 2nd International Workshop on Database Technology and Applications

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“…We also investigated the parabolic optimal control problems by mixed finite element methods, see [36,11]. Very recently, in [29], in order to increase the accuracy of finite element approximations for optimal control problems, they studied two numerical approaches of higher accuracy, namely, the Richardson extrapolation schemes and an interpolation defect correction method.…”

Section: Introductionmentioning

confidence: 99%

Richardson Extrapolation and Defect Correction of Mixed Finite Element Methods for Elliptic Optimal Control Problems

Chen¹,

Huang²,

Hou³

2012

Journal of the Korean Mathematical Society

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Abstract. In this paper asymptotic error expansions for mixed finite element approximations to a class of second order elliptic optimal control problems are derived under rectangular meshes, and the Richardson extrapolation of two different schemes and interpolation defect correction can be applied to increase the accuracy of the approximations. As a byproduct, we illustrate that all the approximations of higher accuracy can be used to form a class of a posteriori error estimators of the mixed finite element method for optimal control problems.

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“…The Richardson extrapolation algorithm has been used in many contexts; see for instance [29,75,77,88,89,121], to mention just some of the contributions.…”

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confidence: 99%

Aproximación de ecuaciones diferenciales mediante una nueva técnica variacional y aplicaciones

Almansa¹,

José²

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ResumenEn esta Tesis presentamos el estudio teórico y numérico de sistemas de ecuaciones diferenciales basado en el análisis de un funcional asociado de forma natural al problema original. Probamos que cuando se utiliza métodos del descenso para minimizar dicho funcional, el algoritmo decrece el error hasta obtener la convergencia dada la no existencia de mínimos locales diferentes a la solución original. En cierto sentido el algoritmo puede considerarse un método tipo Newton globalmente convergente al estar basado en una linearización del problema. Se han estudiado la aproximación de ecuaciones diferenciales rígidas, de ecuaciones rígidas con retardo, de ecuaciones algebraico-diferenciales y de problemas hamiltonianos. Esperamos que esta nueva técnica variacional pueda usarse en otro tipo de problemas diferenciales. AbstractThis thesis is devoted to the study and approximation of systems of differential equations based on an analysis of a certain error functional associated, in a natural way, with the original problem. We prove that in seeking to minimize the error by using standard descent schemes, the procedure can never get stuck in local minima, but will always and steadily decrease the error until getting to the original solution. One main step in the procedure relies on a very particular linearization of the problem, in some sense it is like a globally convergent Newton type method. We concentrate on the approximation of stiff systems of ODEs, DDEs, DAEs and Hamiltonian systems. In all these problems we need to use implicit schemes. We believe that this approach can be used in a systematic way to examine other situations and other types of equations. 10.3.

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Richardson Extrapolation and Defect Correction of Finite Element Methods for Optimal Control Problems

Cited by 5 publications

References 18 publications

Global Superconvergence Estimates of Petrov-Galerkin Methods for Hyperbolic Equations

Global Superconvergence Estimates of Petrov-Galerkin Methods for Hyperbolic Equations

Richardson Extrapolation and Defect Correction of Mixed Finite Element Methods for Elliptic Optimal Control Problems

Aproximación de ecuaciones diferenciales mediante una nueva técnica variacional y aplicaciones

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