Convergence and quasi-optimality of an adaptive finite element method for optimal control problems on \(L^{2}\) errors (Q1685520)

This paper studies the adaptive finite element method for elliptic optimal control problems with integral type control constraints. The main focus is to prove the convergence and quasi-optimality of the adaptive algorithm for all variables including the control, the state and adjoint state in \(L^2\)-norm other than the energy norm. To ensure this, \(H^2\)-regularity for the elliptic equation and mesh grading condition (sufficiently mildly graded) are required. The authors prove the convergence and quasi-optimality of the adaptive algorithm by using the equivalence of the \(L^2\)-norm error and the weighted energy norm errors.