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A posteriori error estimates for optimal control problems governed by parabolic equations

NUMERISCHE MATHEMATIK 2003年第3期93卷 497-521页

作者： Liu, WB Yan, NN Univ Kent CBS Canterbury CT2 7PE Kent England Chinese Acad Sci Acad Math & Syst Sci Inst Syst Sci Beijing Peoples R China

In this paper, we derive a posteriori error estimates for the finite element approximation of quadratic optimal control problem governed by linear parabolic equation. We obtain a posteriori error estimates for both the state and the control approximation. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive finite element approximation schemes for the control problem.

关键词： finite-element approximation Boundary control-problems Variational-inequalities P-laplacian Interpolation

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finite-element approximation of a fourth-order differential equation

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COMPUTERS & MATHEMATICS WITH APPLICATIONS 1998年第8期35卷 95-100页

作者： Shin, JY Pukyong Natl Univ Div Math Sci Nam Gu Pusan 608737 South Korea

A finite-element approximation of a fourth-order differential equation is given. In the direct implementation, a system of nonlinear equations will be obtained and also a full size matrix will be introduced when Newton's method is adopted to solve the system. This difficulty can be avoided by the use of an iterative scheme which is shown to converge to the solution.

关键词： hinged beam fourth-order differential equation finite-element approximation system of nonlinear equations

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Numerical diffusion-induced grain boundary motion

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Interfaces and Free Boundaries 2001年第4期3卷 393-414页

作者： Deckelnick, K. Elliott, C.M. Styles, V. Centre for Mathematical Analysis and its Applications School of Mathematical Sciences University of Sussex Falmer Brighton BN19QH United Kingdom

In this paper we consider the numerical approximation of phase field and sharp interface models for diffusion-induced grain boundary motion. The phase field model consists of a double-obstacle Allen-Cahn equation with a forcing obtained from the solution of a degenerate diffusion equation. On the other hand the sharp interface model consists of forced mean curvature flow coupled to a diffusion equation holding on the interface itself. Formal asymptotics yield the sharp interface model as the limit of the phase field equations as the width of the associated diffuse interface tends to zero. A finite-element approximation of the phase field model is presented and is shown to be convergent to a weak solution. Numerical simulations of both models are described and compared. It is shown that the two models are consistent.

关键词： finite-element approximation Grain boundary motion Mean curvature flow Phase field model Sharp interface

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Numerical approximation of young measures in non-convex variational problems

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NUMERISCHE MATHEMATIK 2000年第3期84卷 395-415页

作者： Carstensen, C Roubícek, T Univ Kiel Math Seminar D-24098 Kiel Germany Charles Univ Inst Math CZ-18675 Prague 8 Czech Republic Acad Sci Czech Republ Inst Informat Theory & Automat CZ-18208 Prague 8 Czech Republic

In non-convex optimisation problems, in particular in non-convex variational problems, there usually does not exist any classical solution but only generalised solutions which involve Young measures. In this paper, first a suitable relaxation and approximation theory is developed together with optimality conditions, and then an adaptive scheme is proposed for the efficient numerical treatment. The Young measures solving the approximate problems are usually composed only from a few atoms. This is the main argument our effective active-set type algorithm is based on. The support of those atoms is estimated from the Weierstrass maximum principle which involves a Hamiltonian whose good guess is obtained by a multilevel technique. Numerical experiments are performed in a one-dimensional variational problem and support efficiency of the algorithm.

关键词： finite-element approximation Microstructure Oscillations Convergence Non-Convex Converge numerical approximation finite element Atoms approximation Theory Microstructure

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Estimates for numerical approximations of rank one convex envelopes

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NUMERISCHE MATHEMATIK 2000年第4期85卷 647-663页

作者： Dolzmann, G Walkington, NJ Carnegie Mellon Univ Dept Math Sci Pittsburgh PA 15213 USA Max Planck Inst Math Sci D-04103 Leipzig Germany

We present a convergence analysis of an algorithm for the numerical computation of the rank-one convex envelope of a function f : M-mxn --> R. A rate of convergence for the scheme is established, and numerical expe... 详细信息

关键词： Nonconvex variational problem finite-element approximation Optimal-design Microstructure Relaxation Converge Optimal design Convexity numerical approximation finite element Numerical Microstructure numerical calculation

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Analysis of optimal control problems for the 2-D stationary Boussinesq equations

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JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 2000年第2期242卷 191-211页

作者： Lee, HC Imanuvilov, OY Ajou Univ Dept Math Suwon 442749 South Korea Iowa State Univ Dept Math Ames IA 50011 USA

This paper deals with optimal control problems associated with the 2-D Boussinesq equations. The controls considered may be of either the distributed or the Neumann type. These problems are first put into an appropriate mathematical formulation. Then the existence of optimal solutions is proved. The use of Lagrange multiplier techniques is justified and an optimality system of equations is derived. (C) 200 Academic Press.

关键词： Navier-stokes equations finite-element approximation Regularity Flow System of equations Navier-Stokes equations Optimal control Flow finite element optimal solution Equation Boussinesq equation

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Convergence results for a nonlinear parabolic control problem

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NUMERICAL FUNCTIONAL ANALYSIS AND OPTIMIZATION 1999年第9-10期20卷 805-824页

作者： Arnautu, V Hömberg, D Sokolowski, J Univ Al I Cuza Fac Math RO-6600 Iasi Romania Weierstrass Inst Appl Anal & Stochast D-10117 Berlin Germany Univ Nancy 1 Inst Elie Cartan Math Lab F-54506 Vandoeuvre Nancy France

We investigate a general control problem for a class of nonlinear parabolic evolution equations. Applications are related to solid-solid and solid-liquid phase transitions. We prove compactness of the solution operator, existence of optimal controls and show convergence of the finite-dimensional approximate control problem to the original one.

关键词： parabolic control problem phase transitions finite-element approximation

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COMPUTABLE ERROR-ESTIMATES FOR LINEARIZATION AND NUMERICAL-SOLUTION OF OBSTACLE PROBLEMS

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JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS 1994年第1期55卷 69-79页

作者： HAN, WM UNIV IOWA DEPT MATHIOWA CITYIA 52242

We derive quantitative a posteriori estimates for the error caused by replacing an obstacle problem with a linear problem. The error bound depends on the solution of the linear problem, but is independent of the solution of the obstacle problem. We then give a discrete version of the a posteriori error estimates, which is used in solving a finite-element system of the obstacle problem. A detailed analysis for a one-dimensional example is given, showing the effectiveness of our error estimates.

关键词： OBSTACLE PROBLEM DUAL VARIATIONAL PRINCIPLE LINEARIZATION finite-element approximation A POSTERIORI ERROR ESTIMATES

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METHOD OF LOCAL approximationS IN THERMOPHYSICAL PROPERTY CALCULATIONS - ESTIMATION OF ERRORS IN DERIVED THERMODYNAMIC PROPERTIES

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INTERNATIONAL JOURNAL OF THERMOPHYSICS 1990年第5期11卷 977-984页

作者： MAGOMEDOV, KM TSENTSIPER, BL ABULAGATOV, IM 1. Institute of Physics of the Dagestan Branch of the USSR Academy of Sciences 367003 Makhachkala USSR

An algorithm for thermophysical property calculations based on the finiteelement approximation in the thermodynamic plane has been suggested. As an example of using the algorithm, errors in obtaining the compressibili... 详细信息

关键词： ethane finite-element approximation P ρ T data thermodynamic plane

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COMPUTERIZED SCHEME FOR THE REACTION OF HEMOGLOBIN WITH LIGANDS

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JOURNAL OF PROTEIN CHEMISTRY 1985年第5期4卷 319-331页

作者： SAMAJA, M NIGGELER, M ROVIDA, E UNIV MILANO DIPARTIMENTO SCI & TECNOL BIOMEDI-20132 MILANOITALY OSPED SAN RAFFAELE CNRCTR STUDI FISIOL LAVORO MUSCOLAREMILANITALY FDN CTR SAN ROMANELLO MONTE TABOR MILANITALY

A scheme for the reaction of hemoglobin with ligands is described, which postulates the functional heterogeneity of the chains, considers all possible combinations of the distribution of the ligand on the four chains of hemoglobin, and does not require simplifying assumptions about the hemoglobin reactivity. Ten tetrameric species are considered, together with 16 reactions between these species, each with an “on” and an “off” rate constant. The dissociation of hemoglobin tetramers into dimers is also considered, with four “on” and four “off” rate constants for the reactions between dimers, and ten equilibrium constants for the reactions between tetramers and dimers. Moreover, some side reactions, such as the “trapping” of ligands by a hemoglobin competitor, are included. A FORTRAN program, suitable for microcomputers, is described for handling this scheme, with some examples showing its advantages.

关键词： hemoglobin computer simulation finite-element approximation kinetic simulation

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