In this paper, we investigate heterogeneous multiscale method （HMM） for the optimal control problem with distributed control constraints governed by elliptic equations with highly oscillatory coefficients. The state...In this paper, we investigate heterogeneous multiscale method （HMM） for the optimal control problem with distributed control constraints governed by elliptic equations with highly oscillatory coefficients. The state variable and co-state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements, whereas the control variable is discretized by the piecewise constant. By applying the well- known Lions＇ Lemma to the discretized optimal control problem, we obtain the necessary and sufficient optimality conditions. A priori error estimates in both L^2 and H^1 norms are derived for the state, co-state and the control variable with uniform bound constants. Finally, numerical examples are presented to illustrate our theoretical results.展开更多
In this paper, we will discuss the asymptotic behaviour for a class of hyperbolic -parabolic type equation with highly oscillatory coefficients arising from the strong-transient heat and mass transfer problems of comp...In this paper, we will discuss the asymptotic behaviour for a class of hyperbolic -parabolic type equation with highly oscillatory coefficients arising from the strong-transient heat and mass transfer problems of composite media. A complete multiscale asymptotic expansion and its rigorous verification will be reported.展开更多
基金The work was supported by the Shandong Province Outstanding Y- oung Scientists Research Award Fund Project （Grant No. BS2013DX010）, by the Natural Science Foundation of Shandong Province, China （Grant No. ZR2011FQ030, ZR2013FQ001, ZR2013FM025）, by Natural Science Foundation of China （Grant No. 11501326 and 11571356）, and by the Shandong Academy of Sciences Youth Fund Project （Grant No. 2013QN007）.
文摘In this paper, we investigate heterogeneous multiscale method （HMM） for the optimal control problem with distributed control constraints governed by elliptic equations with highly oscillatory coefficients. The state variable and co-state variable are approximated by the multiscale discretization scheme that relies on coupled macro and micro finite elements, whereas the control variable is discretized by the piecewise constant. By applying the well- known Lions＇ Lemma to the discretized optimal control problem, we obtain the necessary and sufficient optimality conditions. A priori error estimates in both L^2 and H^1 norms are derived for the state, co-state and the control variable with uniform bound constants. Finally, numerical examples are presented to illustrate our theoretical results.
基金This work is Supported by National Natural Science Foundation of China ( No. 19801006) Special Funds for Major State Basic Research Projects ( No. G2000067102).
文摘In this paper, we will discuss the asymptotic behaviour for a class of hyperbolic -parabolic type equation with highly oscillatory coefficients arising from the strong-transient heat and mass transfer problems of composite media. A complete multiscale asymptotic expansion and its rigorous verification will be reported.
