An unfitted HDG method for a distributed optimal convection-diffusion control problem
This work addresses computational challenges in distributed optimal control for convection-diffusion problems with complex geometries, but it is incremental as it builds on existing HDG and unfitted methods.
The paper tackles an optimal control problem governed by a convection-diffusion equation in a domain with a non-fitting computational mesh, using an unfitted HDG method and Transfer Path Method to handle boundary data transfer. It proves optimal convergence rates in the L^2-norm for state and adjoint variables and provides numerical validation.
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 $\partial Ω$. The computational domain $Ω_h$ does not necessarily fit $Ω$ and the Transfer Path Method (TPM) is used to transfer the boundary data from $\partial Ω$ to $\partial Ω_h$ through segments of direction $\boldsymbol{m}$. Under closeness conditions between $\partial Ω_h$ and $\partial Ω$ 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.