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New a posteriori \(L^\infty(L^2)\) and \(L^2(L^2)\)-error estimates of mixed finite element methods for general nonlinear parabolic optimal control problems. (English) Zbl 1389.49018
New a-posteriori error estimates are derived for nonlinear parabolic optimal control problems with objective functionals containing the gradient of the state variable. The author uses the mixed finite element for discretization of the state and adjoint equations with which both the scalar variable and its flux variable are approximated in the same accuracy. The control is approximated by piecewise constant functions. The nonlinear parabolic optimal control problem is discretized in time by the backward Euler method and the discretized optimal control problem is solved using the preconditioned projection gradient method.
The a-posteriori error estimates are used for developing reliable adaptive mixed finite element approximation for the optimal control problem. The performance of the a-posteriori error estimators with the adaptive finite element is demonstrated by two numerical examples. The numerical results show that the theoretical results and the approximations are obviously more efficient than on uniform grids.
MSC:
49K20 Optimality conditions for problems involving partial differential equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
49M30 Other numerical methods in calculus of variations (MSC2010)
49M25 Discrete approximations in optimal control
35K55 Nonlinear parabolic equations
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