A Nitsche-eXtended finite element method for distributed optimal control problems of elliptic interface equations

TL;DR

This paper introduces a Nitsche-eXtended finite element method for solving distributed optimal control problems governed by elliptic interface equations, providing error estimates and numerical validation.

Contribution

It develops a novel interface-unfitted finite element approach using Nitsche's method with enriched basis functions for better accuracy in interface problems.

Findings

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

Numerical results confirm the theoretical error bounds.

The method effectively handles interface-unfitted discretizations.

Abstract

This paper analyzes an interface-unfitted numerical method for distributed optimal control problems governed by elliptic interface equations. We follow the variational discretization concept to discretize the optimal control problems, and apply a Nitsche-eXtended finite element method to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around the interface are enriched into the standard linear element space. Optimal error estimates of the state, co-state and control in a mesh-dependent norm and the $L^2$ norm are derived. Numerical results are provided to verify the theoretical results.