A priori and a posteriori error estimates of H¹‐Galerkin mixed finite element method for parabolic optimal control problems

Abstract

Summary In this exposition, we study both a priori and a posteriori error analysis for the H1‐Galerkin mixed finite element method for optimal control problems governed by linear parabolic equations. The state and costate variables are approximated by the lowest order Raviart‐Thomas finite element spaces, whereas the control variable is approximated by piecewise constant functions. Compared to the standard mixed finite element procedure, the present method is not subject to the Ladyzhenskaya‐Babuska‐Brezzi condition and the approximating finite element spaces are allowed to be of different degree polynomials. A priori error analysis for both the semidiscrete and the backward Euler fully discrete schemes are analyzed, and L∞false(L2false) convergence properties for the state variables and the control variable are obtained. In addition, L2(L2)‐norm a posteriori error estimates for the state and control variables and L∞false(L2false)‐norm for the flux variable are also derived.

Guaranteed lower bounds for cost functionals of time-periodic parabolic optimization problems

“…Functional a posteriori estimation provides a useful machinery to derive computable and guaranteed quantities for the desired unknown solution, see, e.g., [39,12] on parabolic problems. Recent works on new estimates for parabolic problems and parabolic optimal control problems can be found in [32] and [42], respectively. A posteriori estimates of functional type for elliptic optimal control problems can be found in [10,11,40,31].…”