An Interface-Unfitted Finite Element Method for Elliptic Interface Optimal Control Problems

Abstract

This paper develops and analyses numerical approximation for linear-quadratic optimal control problems governed by elliptic interface equations. We adopt variational discretization concept to discretize optimal control problems, and apply an interface-unfitted finite element method due to [A. Hansbo and P. Hansbo. An unfitted finite element method, based on Nitsche's method, for elliptic interface problems. Comput. Methods Appl. Mech. Engrg., 191(47-48): 5537-5552, 2002] to discretize the corresponding state and adjoint equations, where piecewise cut basis functions around interface are enriched into standard conforming finite element space. Optimal error estimates in both $L$² norm and a mesh-dependent norm are derived for the optimal state, co-state and control under different regularity assumptions. Numerical results verify the theoretical results.