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This article discusses the quasi-optimality of adaptive nonconforming finite element methods for distributed optimal control problems governed by -harmonic operators form=1,2m=1,2. A variational discretization approach is employed and the state and adjoint variables are discretized using nonconforming finite elements. Error equivalence results at the continuous and discrete levels lead to a priori and a posteriori error estimates for the optimal control problem. The general axiomatic framework that includes stability, reduction, discrete reliability, and quasi-orthogonality establishes the quasi-optimality. Numerical results demonstrate the theoretically predicted orders of convergence and the efficiency of the adaptive estimator.

Convergence of Adaptive Crouzeix–Raviart and Morley FEM for Distributed Optimal Control Problems