An unfitted HDG method for a distributed optimal convection-diffusion control problem

TL;DR

This paper presents an analysis of a high-order unfitted HDG method for an optimal control problem governed by a convection-diffusion equation, utilizing the Transfer Path Method to handle domain boundary discrepancies.

Contribution

It introduces a novel unfitted HDG approach with theoretical convergence guarantees for complex domains using the Transfer Path Method.

Findings

Achieved optimal convergence rates in the $L^2$-norm for all variables.

Validated theoretical results with numerical experiments.

Handled complex domain boundaries without mesh fitting.

Abstract

We analyze a high order unfitted hybridizable discontinuous Galerkin (HDG) method for an optimal control problem governed by a convection-diffusion equation posed in a domain with piecewise-wise $\mathcal{C}^2$ boundary $\partial \Omega$. The computational domain $\Omega_h$ does not necessarily fit $\Omega$ and the Transfer Path Method (TPM) is used to transfer the boundary data from $\partial \Omega$ to $\partial \Omega_h$ through segments of direction $\boldsymbol{m}$. Under closeness conditions between $\partial \Omega_h$ and $\partial \Omega$ and on the transfer vector $\boldsymbol{m}$, we prove optimal order of convergence in the $L^2$-norm for all variables of the state and adjoint problems. We also show numerical examples to complement the theory.