A non-overlapping DDM combined with the characteristic method for optimal control problems governed by convection-diffusion equations
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 discrete scheme is constructed (Chapter 3.1). Thereby the authors utilize backward difference quotients, and along the inter-domain boundaries jump relations, integral mean values and flux calculations. In Chapter 3.2, the projection method is applied and a projection algorithm is presented in order to solve the scheme from Chapter 3.1. Chapter 4 deals with convergence analysis and error estimates for \(u-U,y-Y(u),p-P(u)\) (\(\{y,u,p\}\) the exact solution, \(\{Y,U,P\}\) the approximation solution) can be proved (Theorem 4.9) and the authors emphasize that a time set constraint is needed, comparable as in references. Finally, Chapter 5 gives a lot of numerical experiments and some beautiful figures comparing exact and approximated solutions for the states and controls.
scientific article; zbMATH DE number 6946585
scientific article