A Nitsche-eXtended finite element method for distributed optimal control problems of elliptic interface equations
This work provides a rigorous numerical analysis for interface-unfitted methods in optimal control, which is valuable for problems with complex interfaces but represents an incremental extension of existing techniques.
The paper develops and analyzes a Nitsche-eXtended finite element method for distributed optimal control problems governed by elliptic interface equations, achieving optimal error estimates for state, co-state, and control in mesh-dependent and L2 norms, verified by numerical experiments.
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.