A posteriori $L^{infty}(L^{2})$-error estimates of semidiscrete mixed finite element methods for hyperbolic optimal control problems

摘要

In this paper, we discuss the a posteriori error estimates of the semidiscrete mixed finite element methods for quadratic optimal control problems governed by linear hyperbolic equations. The state and the co-state are discretized by the order $k$ Raviart-Thomas mixed finite element spaces and the control is approximated by piecewise polynomials of order $k(kgeq0)$. Using mixed elliptic reconstruction method, a posteriori $L^{infty}(L^{2})$-error estimates for both the state and the control approximation are derived. Such estimates, which are apparently not available in the literature, are an important step towards developing reliable adaptive mixed finite element approximation schemes for the control problem.