We revisit the L 2 norm error estimate for the C 0 interior penalty analysis of fourth order Dirichlet boundary control problem. The L 2 norm estimate for the optimal control is derived under reduced regularity assumption and this analysis can be carried out on any convex polygonal domains. Residual based a-posteriori error bounds are derived for optimal control, state and adjoint state variables under minimal regularity assumptions. The estimators are shown to be reliable and locally efficient. The theoretical findings are illustrated by numerical experiments.

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The main aim of this article is to analyze mixed finite element method for the second order Dirichlet boundary control problem. Therein, we develop both a priori and a posteriori error analysis using the energy space based approach. We obtain optimal order a priori error estimates in the energy norm and L 2 -norm with the help of auxiliary problems. The reliability and the efficiency of proposed a posteriori error estimator is discussed using the Helmholtz decomposition. Numerical experiments are presented to confirm the theoretical findings.