A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection-diffusion equations
type: Item
instance of: scholarly article
title: A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection-diffusion equations
zbMATH Open document ID: 1410.90242
author: Keying Ma
author: Tongjun Sun
published in: Computational Optimization and Applications
publication date: 2018-10-02T00:00:00Z
review text: The authors extend their research results on the numerical treatment of convection-diffusion equations (non-overlapping decomposition of the domain and use of the method of characteristics) to the task of treating an optimal control problem, whereby the associated partial differential equations are just those mentioned above. In order to prepare the numerical treatment, the results achieved so far are adapted first, whereby the transition conditions along the inter-domain boundaries between the subdomains, the use of characteristics and of methodologies of optimal control play an important role. Then follows the main part of the work: proving a priori error estimates for the state, co-state and control variables. Explicit numerical calculation results correspond to the obtained estimates. More exactly, the considered control problem has the form \[ \min_{u\in K}\{\frac{1}{2} \int_0^T (\|y-z_d \|^2 + \alpha \|u\|^2)dt\} \] with \(z_d(x,t) \in H^1(0,T;L^2(\Omega))\) a known target function, with the state \(y(x,t) \in W = L^2(0,T;V)\), \(V=H_0^1(\Omega)\), \(\Omega\) a bounded convex polygon domain in \(\mathbb R^2\) and with the control \(u(x,t) \in K= \{u\in X:u\geq 0 \; \text{a.e.} \; \text{in} \;\Omega \times (0,T] \} \) satisfying in \(\Omega \times (0,T]\) the differential equation (\(b\) is a velocity field in the flow control, \(B\) a linear continuous operator from \(L^2(0,T;U) \) to \(L^2(0,T;V')\), \(A\) a uniformly positive definite matrix componentwisely in \(W^{1,\infty}(\overline{\Omega})\) \[ \frac{\partial y}{\partial t} + b.\nabla y - \nabla.(A\nabla y) = f + Bu \] with boundary and initial conditions (and incompressible flow). The weak form of that problem and optimal control theory gives the optimality system (for the triplet \((y,u,p)\), \(p \in H^1(0,T;L^2(\Omega))\) is the co-state), which is to solve numerically. For this aim, using a non-overlapping domain decomposition (DDM) together with the characteristic method, a fully parallel and discre…
reviewed by: Alfred Göpfert
Mathematics Subject Classification ID: 90C48
Mathematics Subject Classification ID: 90C59
Mathematics Subject Classification ID: 65M25
Mathematics Subject Classification ID: 65N55
Mathematics Subject Classification ID: 49J20
Mathematics Subject Classification ID: 49M05
zbMATH DE Number: 6946585
zbMATH Keywords: convection-diffusion equations
zbMATH Keywords: optimal control problems
zbMATH Keywords: non-overlapping DDM
zbMATH Keywords: integral mean method
zbMATH Keywords: error estimates
MaRDI profile type: Publication
full work available at URL: s10589-018-0008-0
OpenAlex ID: W2804812437
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Wikidata QID: Q129798536
DOI: 10.1007/S10589-018-0008-0
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description: scientific article; zbMATH DE number 6946585