On $C^0$ Interior Penalty Method for Fourth Order Dirichlet Boundary Control Problem and a New Error Analysis for Fourth Order Elliptic Equation with Cahn-Hilliard Boundary Condition

In this paper, we revisit the $L_2$-norm error estimate for $C^0$-interior penalty analysis of Dirichlet boundary control problem governed by biharmonic operator. In this work, we have relaxed the interior angle condition of the domain from $120$ degrees to $180$ degrees, therefore this analysis can be carried out for any convex domain. The theoretical findings are illustrated by numerical experiments. Moreover, we propose a new analysis to derive the error estimates for the biharmonic equation with Cahn-Hilliard type boundary condition under minimal regularity assumption.