An interface-unfitted finite element method for elliptic interface optimal control problem

TL;DR

This paper introduces an interface-unfitted finite element method for elliptic interface optimal control problems, providing theoretical error estimates and numerical validation for improved accuracy around interfaces.

Contribution

It develops a novel interface-unfitted finite element approach for elliptic interface optimal control problems with rigorous error analysis and numerical verification.

Findings

Optimal error estimates in $L^2$ norm and mesh-dependent norm.

Numerical results confirm theoretical error bounds.

Enrichment of basis functions improves interface approximation.

Abstract

This paper develops and analyses numerical approximation for linear-quadratic optimal control problem governed by elliptic interface equations. We adopt variational discretization concept to discretize optimal control problem, 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 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 optimal state, co-state and control under different regularity assumptions. Numerical results verify the theoretical results.