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

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 $\partial \hat{I}\text{c◯}$. The computational domain $\hat{I}{\text{c◯}}_h$ does not necessarily fit $\hat{I}\text{c◯}$ and the Transfer Path Method (TPM) is used to transfer the boundary data from $\partial \hat{I}\text{c◯}$ to $\partial \hat{I}{\text{c◯}}_h$ through segments of direction $\mathbf{m}$. Under closeness conditions between $\partial \hat{I}{\text{c◯}}_h$ and $\partial \hat{I}\text{c◯}$ and on the transfer vector $\mathbf{m}$, 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.