A Neumann interface optimal control problem with elliptic PDE constraints and its discretization and numerical analysis

TL;DR

This paper investigates an elliptic PDE optimal control problem with interface conditions, proposing a discretization method using immersed finite elements and postprocessing to improve accuracy, supported by theoretical error estimates and numerical validation.

Contribution

It introduces a novel discretization approach for interface control problems using immersed finite elements and superconvergence postprocessing, with comprehensive error analysis.

Findings

Optimal error estimates for control achieved

Superconvergence in control approximation demonstrated

Numerical results confirm theoretical accuracy improvements

Abstract

We study an optimal control problem governed by elliptic PDEs with interface, which the control acts on the interface. Due to the jump of the coefficient across the interface and the control acting on the interface, the regularity of solution of the control problem is limited on the whole domain, but smoother on subdomains. The control function with pointwise inequality constraints is served as the flux jump condition which we called Neumann interface control. We use a simple uniform mesh that is independent of the interface. The standard linear finite element method can not achieve optimal convergence when the uniform mesh is used. Therefore the state and adjoint state equations are discretized by piecewise linear immersed finite element method (IFEM). While the accuracy of the piecewise constant approximation of the optimal control on the interface is improved by a postprocessing step which possesses superconvergence properties; as well as the variational discretization concept for the optimal control is used to improve the error estimates. Optimal error estimates for the control, suboptimal error estimates for state and adjoint state are derived. Numerical examples with and without constraints are provided to illustrate the effectiveness of the proposed scheme and correctness of the theoretical analysis.