Geodesic
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
Applications of Mathematics, Tome 61 (2016) no. 2, pp. 135-163.

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We study new a posteriori error estimates of the mixed finite element methods for general optimal control problems governed by nonlinear parabolic equations. The state and the co-state are discretized by the high order Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise constant functions. We derive a posteriori error estimates in $L^{\infty }(J;L^2(\Omega )) $-norm and $L^2(J;L^2(\Omega ))$-norm for both the state, the co-state and the control approximation. Such estimates, which seem to be new, are an important step towards developing a reliable adaptive mixed finite element approximation for optimal control problems. Finally, the performance of the posteriori error estimators is assessed by two numerical examples.
DOI : 10.1007/s10492-016-0126-x
Classification : 49J20, 65N30
Keywords: a posteriori error estimate; general optimal control problem; nonlinear parabolic equation; mixed finite element method 
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Lu, Zuliang. 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. Applications of Mathematics, Tome 61 (2016) no. 2, pp. 135-163. doi : 10.1007/s10492-016-0126-x. http://geodesic.mathdoc.fr/articles/10.1007/s10492-016-0126-x/

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