A Frame Work for the Error Analysis of Discontinuous Finite Element Methods for Elliptic Optimal Control Problems and Applications to $C^0$ IP methods
This provides a unified error analysis framework for discontinuous finite element methods in optimal control, benefiting researchers working on numerical methods for PDE-constrained optimization.
The paper develops an abstract framework for error analysis of discontinuous Galerkin methods for control-constrained optimal control problems, achieving best approximation results and reliable a posteriori error estimators. Applications to C0 interior penalty methods for biharmonic equation control problems are validated with numerical experiments.
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.