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

Abstract

Available online xxxx Keywords:Multiscale control problem Mixed finite element Homogenization A prior error estimates a b s t r a c tWe 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.

Generalized Rough Polyharmonic Splines for Multiscale PDEs with Rough Coefficients

“…There has been many existing work concerning the design of novel numerical methods for multiscale problems and the mathematics to foresee and assess their performance in engineering and scientific applications, such as homogenization [25,41], numerical homogenization [2,13,44], heterogeneous multi-scale methods [1,14,27,30], multiscale network approximations [6], multi-scale finite element methods [3,8,15,17], variational multi-scale methods [5,24], flux norm homogenization [7,38], rough polyharmonic splines (RPS) [40], generalized multi-scale finite element methods [10,11,16], localized orthogonal decomposition [21,22,29,42], etc. Fundamental questions for numerical homogenization are: how to approximate the high dimensional solution space by a low dimensional approximation space with optimal error control, and furthermore, how to construct the approximation space efficiently, for example, whether its basis can be localized on a coarse patch.…”

Finite Element Approximation of Optimal Control Problem with Weighted Extended B-Splines

“…Hou also discussed the error estimates of expanded mixed FEM for OCPs obtained by hyperbolic integro differential equation [26]. Chen et al [27] applied mixed multiscale FEM for the convex OCPs with oscillating coefficients. Error estimates for mixed FEM for OCPs of low regularity are discussed in [28].…”

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.…”