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Crank–Nicolson finite element methods using symmetric stabilization with an application to optimal control problems subject to transient advection–diffusion equations. (English) Zbl 1279.49019
Summary: We consider a finite element method with symmetric stabilization for transient advection-diffusion-reaction problems. The Crank–Nicolson finite difference scheme is used for discretization in time. We prove stability of the numerical method both for implicit and explicit treatment of the stabilization operator. The resulting convergence results are given and the results are illustrated by a numerical experiment. We then consider a model problem for PDE-constrained optimization. Using discrete adjoint consistency of our stabilized method, we show that both the implicit and semi-implicit methods proposed yield optimal convergence for the control and the state variable.

MSC:
49M25 Discrete approximations in optimal control
49M29 Numerical methods involving duality
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M38 Boundary element methods for initial value and initial-boundary value problems involving PDEs
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