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A nonconforming finite element method for constrained optimal control problems governed by parabolic equations. (English) Zbl 1448.65161

Summary: In this paper, a nonconforming finite element method (NFEM) is proposed for the constrained optimal control problems (OCPs) governed by parabolic equations. The time discretization is based on the finite difference methods. The state and co-state variables are approximated by the nonconforming \(EQ_1^{\operatorname{rot}}\) elements, and the control variable is approximated by the piecewise constant element, respectively. Some superclose properties are obtained for the above three variables. Moreover, for the state and co-state, the convergence and superconvergence results are achieved in \(L^2\)-norm and the broken energy norm, respectively.

MSC:

65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
49M25 Discrete approximations in optimal control
49J20 Existence theories for optimal control problems involving partial differential equations
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References:

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