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

TL;DR

This paper advances the analysis of $C^{0}$ interior penalty methods for fourth-order boundary control problems, extending applicability to convex domains with relaxed angle conditions and providing new error estimates for related elliptic equations.

Contribution

It introduces a relaxed interior angle condition for convex domains and a novel error analysis for biharmonic equations with Cahn-Hilliard boundary conditions under minimal regularity.

Findings

Error estimates are validated through numerical experiments.

The analysis applies to convex domains with interior angles up to 180 degrees.

New error bounds are derived for biharmonic equations with Cahn-Hilliard boundary conditions.

Abstract

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.