A mixed multiscale finite element method for convex optimal control problems with oscillating coefficients. (August 2015)

• Record Type: Journal Article
• Authors: Chen, Yanping; Huang, Yunqing; Liu, Wenbin; Yan, Ningning
• Is Part Of: Computers & mathematics with applications. Volume 70:issue 4(2015)
• Page Start: 297
• Page End: 313
• Publication Date: 2015-08
• Subjects: Multiscale control problem -- Mixed finite element -- Homogenization -- A prior error estimates; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2015.03.020
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: We study numerical approximation of convex optimal control problems governed by elliptic partial differential equations with oscillating coefficients. Since the objective functional contains flux, we approximate the problems using the mixed finite element methods. We first analyze the standard mixed finite element approximation schemes. Then, motivated by the numerical simulation of the primal variable and the flux in highly heterogeneous porous media, we use a multiscale mixed finite element method to solve the state equations. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. The analysis of the approximate control problems is carried out under the assumption that the oscillating coefficients are locally periodic, which allows us to use homogenization theory to obtain the asymptotic structure of the solutions, although the numerical schemes are designed for the general case.