A space-time interface-fitted method for moving-subdomain distributed control problems with energy regularization

TL;DR

This paper introduces a novel space-time interface-fitted Petrov-Galerkin method for moving-interface optimal control problems with energy regularization, providing error estimates and numerical validation.

Contribution

It develops a new unstructured mesh-based approximation for moving-interface control problems, including error analysis and numerical experiments.

Findings

Optimal error estimates are established under regularity assumptions.

Numerical results confirm the theoretical error bounds.

The method effectively handles moving interfaces in control problems.

Abstract

This paper investigates a space-time interface-fitted approximation of a moving-interface optimal control problem with energy regularization. We reformulate the optimality conditions into a variational problem involving both the state and adjoint. This problem is shown to be equivalent to our optimal control problem. Based on fully unstructured, space-time interface-fitted meshes, we propose and analyze a Petrov-Galerkin approximation of the problem. An optimal error estimate with respect to a discrete norm is established under a specific regularity assumption on the state and adjoint. Several numerical results are presented to corroborate our theoretical results.