A posteriori error estimates for hp finite element solutions of convex optimal control problems

In this paper, we present a posteriori error analysis for $hp$ finite element approximation of convex optimal control problems. We derive a new quasi‐interpolation operator of Clément type and a new quasi‐interpolation operator of Scott–Zhang type that preserves homogeneous boundary condition. The Scott–Zhang type quasi‐interpolation is suitable for an application in bounding the errors in $L^2$ ‐norm. Then $hp$ a posteriori error estimators are obtained for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive finite elements for the control problems.