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A priori error estimates of finite volume method for nonlinear optimal control problem. (Russian, English) Zbl 1399.65142
Sib. Zh. Vychisl. Mat. 20, No. 3, 273-288 (2017); translation in Numer. Analysis Appl. 10, No. 3, 224-236 (2017).
Summary: In this paper, we study a priori error estimates for the finite volume element approximation of nonlinear optimal control problem. The schemes use discretizations based on a finite volume method. For the variational inequality, we use the method of the variational discretization concept to obtain the control. Under some reasonable assumptions, we obtain some optimal order error estimates. The approximate order for the state, costate and control variables is \(O(h^2)\) or \(O(h^2\sqrt{| \ln h|})\) in the sense of \(L^2\)-norm or \(L^{\infty}\)-norm. A numerical experiment is presented to test the theoretical results. Finally, we give some conclusions and future works.
65K10 Numerical optimization and variational techniques
65N08 Finite volume methods for boundary value problems involving PDEs
49J20 Existence theories for optimal control problems involving partial differential equations
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