Extended finite element methods for optimal control problems governed by Poisson equation in non-convex domains

TL;DR

This paper develops and analyzes two XFEM approaches for solving optimal control problems governed by Poisson equations in non-convex domains, providing error estimates and numerical validation.

Contribution

It introduces two XFEM discretization methods for these problems and derives optimal error estimates, advancing numerical techniques for non-convex domain control problems.

Findings

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

Numerical results confirm theoretical error bounds.

Two XFEM methods effectively handle non-convex domain complexities.

Abstract

This paper analyzes two eXtended finite element methods (XFEMs) for linear quadratic optimal control problems governed by Poisson equation in non-convex domains. We follow the variational discretization concept to discretize the continuous problems, and apply an XFEM with a cut-off function and a classic XFEM with a fixed enrichment area to discretize the state and co-state equations. Optimal error estimates are derived for the state, co-state and control. Numerical results confirm our theoretical results.