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Superconvergence of Triangular Mixed Finite Elements Solutions for Optimal Control Problems

Author: ZhangJuan
Tutor: ChenYanPing
School: Xiangtan University
Course: Computational Mathematics
Keywords: Mixed Finite Element Optimal Control Superconvergence Error estimates Triangulation Post-processing operator
CLC: O241.82
Type: Master's thesis
Year: 2008
Downloads: 34
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Abstract


Effective numerical methods is key to the optimal control problem is widely applied to various practice areas, these numerical methods, although there are other methods being used, but the finite element method is the most common application of a theoretical analysis and algorithm implementation has been maturing. Currently, the article on the theoretical analysis of mixed finite element approximation of optimal control problems and do not have much. This article, the target functional optimal control problems with state variable gradient, in order to improve the accuracy of the gradient, we use triangular mixed finite element method to discretize, namely the state variables (scalar and vector) and the control variables take different finite element space approximation: the state variable and its gradient, we used RaViart-Thomas mixed finite element space approximation; piecewise constant space approximation of the control variables. First, the use of the variational principle of optimality conditions for optimal control problems, the upcoming seeking a minimal functional problem is transformed into the equation of state, accompanied by the equation of state and a variational inequality three simultaneous system. Then we define the control variable interpolation u I Inequalities through the use of the change in optimal conditions, estimates of the control variables will be converted to dual state quantities estimated. Control centralized function, we introduced some associated therewith and as a state function of the intermediate variable and the accompanying state function, the finite element error is divided into several parts to be estimated proved interpolation the U I and The estimated between discrete solution u h h 2 SUPERCONVERGENCE. Post-processing for the control variable low positive then the sexual introduced special after processing operator sub because after processing operator introduced dominating set of related, different control sets are to be the introduction of the post-processing operator child is different. The control set U ad = {u ∈ L 2 (Ω): the ∫ Ω u (x) dx ≥ 0}, therefore We introduce the corresponding post-processing operator U- * = g (h) = (max (0, (?) h )-Z h < / sub>) / v. The estimated processing operator and then use the same method to get the exact solution h 2 Superconvergence nature. Finally, we will give two numerical examples to verify Superconvergence phenomenon.

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CLC: > Mathematical sciences and chemical > Mathematics > Computational Mathematics > Numerical Analysis > The numerical solution of differential equations, integral equations > Numerical Solution of Partial Differential Equations
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