Nitsche-XFEM for optimal control problems governed by elliptic PDEs with interfaces

TL;DR

This paper introduces a Nitsche-XFEM-based numerical method for solving optimal control problems governed by elliptic PDEs with interfaces, providing optimal error estimates and applicability to non-homogeneous interface conditions.

Contribution

The paper develops a novel Nitsche-XFEM approach for interface PDEs in optimal control, with proven optimal error estimates and flexibility for non-homogeneous conditions.

Findings

Optimal error estimates for state, co-state, and control.

Method is effective for non-homogeneous interface conditions.

Numerical results confirm theoretical accuracy.

Abstract

For the optimal control problem governed by elliptic equations with interfaces, we present a numerical method based on the Hansbo's Nitsche-XFEM. We followed the Hinze's variational discretization concept to discretize the continuous problem on a uniform mesh. We derive optimal error estimates of the state, co-state and control both in mesh dependent norm and L2 norm. In addition, our method is suitable for the model with non-homogeneous interface condition. Numerical results confirmed our theoretical results, with the implementation details discussed.