Esteban Henríquez

Esteban Henríquez, Jeonghun J. Lee, Sander Rhebergen

We present parameter-robust preconditioners for linear systems that arise after applying static condensation to a hybridizable discontinuous Galerkin (HDG) discretization of the time-dependent Stokes problem. Building upon the theoretical framework introduced in our previous work [SIAM Journal on Scientific Computing, 47(6):A3212-A3238, 2025], we extend the analysis to derive new preconditioners that remain robust with respect to all physical and discretization parameters. The construction relies on first establishing uniform well-posedness of the HDG formulation (before static condensation) through appropriately defined norms. Based on this result, we identify sufficient conditions that a norm on the face space must satisfy to guarantee parameter-robustness of the resulting preconditioner for the statically condensed HDG system. Numerical experiments in two and three dimensions verify our theoretical results.

Esteban Henríquez, Manuel Solano

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.