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

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