A Frame Work for the Error Analysis of Discontinuous Finite Element Methods for Elliptic Optimal Control Problems and Applications to $C^0$ IP methods

TL;DR

This paper develops an abstract error analysis framework for discontinuous Galerkin methods applied to elliptic optimal control problems, including $C^0$ interior penalty methods, supported by numerical experiments.

Contribution

It introduces a unified error analysis framework for discontinuous Galerkin methods in control problems, applicable under minimal regularity, and extends to $C^0$ interior penalty and variational discretizations.

Findings

Establishes best approximation results and reliable a posteriori error estimators.

Demonstrates applicability to boundary and distributed control problems governed by biharmonic equations.

Numerical experiments confirm theoretical error estimates.

Abstract

In this article, an abstract framework for the error analysis of discontinuous Galerkin methods for control constrained optimal control problems is developed. The analysis establishes the best approximation result from a priori analysis point of view and delivers reliable and efficient a posteriori error estimators. The results are applicable to a variety of problems just under the minimal regularity possessed by the well-posed ness of the problem. Subsequently, applications of $C^0$ interior penalty methods for a boundary control problem as well as a distributed control problem governed by the biharmonic equation subject to simply supported boundary conditions are discussed through the abstract analysis. Numerical experiments illustrate the theoretical findings. Finally, we also discuss the variational discontinuous discretization method (without discretizing the control) and its corresponding error estimates.