Numerical Methods for Constrained Elliptic Optimal Control Problems with Rapidly Oscillating Coefficients

Abstract

In this paper we use two numerical methods to solve constrained optimal control problems governed by elliptic equations with rapidly oscillating coefficients: one is finite element method and the other is multiscale finite element method. We derive the convergence analysis for those two methods. Analytical results show that finite element method can not work when the parameter ε is small enough, while multiscale finite element method is useful for any parameter ε.

A mixed multiscale finite element method for convex optimal control problems with oscillating coefficients

“…Such problems occur in applications involving control of water injection into oil-wells, and properties design of composite materials. Our recent work (see [20]) serves as a contribution to the development of the multiscale finite element method for solving optimal control problems by elliptic partial differential equations with oscillating coefficients.…”

Optimal control for multiscale equations with rough coefficients

“…Numerical homogenization for problems with multiple scales have attracted increasing attention in recent years. If the coefficient a(x) has structural properties such as scale separation and periodicity, together with some regularity assumptions (e.g., a(x) ∈ W 1,∞ ), classical homogenization [29,26] can be used to construct efficient multiscale computational methods and have been applied to optimal control problems, such as multiscale asymptotic expansions method [6,7,30], multiscale finite element method (MsFEM) [24,12,10,11], and heterogeneous multiscale method (HMM) [42,21].…”

“…In the context of optimal control, homogenization based methods have been applied to problems governed by multiscale PDEs with separable scales [10,30,11]. To the best of our knowledge, few literature concerns the optimal control with nonseparable scales, which is of great importance for applications.…”

“…Multiscale numerical methods such as multiscale finite element method (MsFEM) [24,25] can be used to efficiently capture the large scale components of the solution on the coarse grid. In [11,Theorem 4.11] and [27], MsFEM was applied to solve the control problem governed by multiscale PDEs with separable scales, a priori error estimates can be obtained as follows,…”