Sander Rhebergen

Sander Rhebergen, Garth N. Wells

We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier--Stokes equations for which the approximate velocity field is pointwise divergence-free. The method builds on the method presented by Labeur and Wells [SIAM J. Sci. Comput., vol. 34 (2012), pp. A889--A913]. We show that with modifications of the function spaces in the method of Labeur and Wells it is possible to formulate a simple method with pointwise divergence-free velocity fields which is momentum conserving, energy stable, and pressure-robust. Theoretical results are supported by two- and three-dimensional numerical examples and for different orders of polynomial approximation.

Sander Rhebergen, Garth N. Wells

Stability and error analysis of a hybridized discontinuous Galerkin finite element method for Stokes equations is presented. The method is locally conservative, and for particular choices of spaces the velocity field is point-wise solenoidal. It is shown that the method is inf-sup stable for both equal-order and locally Taylor--Hood type spaces, and \emph{a priori} error estimates are developed. The considered method can be constructed to have the same global algebraic structure as a conforming Galerkin method, unlike standard discontinuous Galerkin methods that have greater number of degrees of freedom than conforming Galerkin methods on a given mesh. We assert that this method is amongst the simplest and most flexible finite element approaches for Stokes flow that provide local mass conservation. With this contribution the mathematical basis is established, and this supports the performance of the method that has been observed experimentally in other works.

Sander Rhebergen, Garth N. Wells

We present optimal preconditioners for a recently introduced hybridized discontinuous Galerkin finite element discretization of the Stokes equations. Typical of hybridized discontinuous Galerkin methods, the method has degrees-of-freedom that can be eliminated locally (cell-wise), thereby significantly reducing the size of the global problem. Although the linear system becomes more complex to analyze after static condensation of these element degrees-of-freedom, the pressure Schur complement of the original and reduced problem are the same. Using this fact, we prove spectral equivalence of this Schur complement to two simple matrices, which is then used to formulate optimal preconditioners for the statically condensed problem. Numerical simulations in two and three spatial dimensions demonstrate the good performance of the proposed preconditioners.

Tamas L. Horvath, Sander Rhebergen

We present a finite element method for the incompressible Navier--Stokes problem that is locally conservative, energy-stable and pressure-robust on time-dependent domains. To achieve this, the space--time formulation of the Navier--Stokes problem is considered. The space--time domain is partitioned into space--time slabs which in turn are partitioned into space--time simplices. A combined discontinuous Galerkin method across space--time slabs, and space--time hybridized discontinuous Galerkin method within a space--time slab, results in an approximate velocity field that is $H({\rm div})$-conforming and exactly divergence-free, even on time-dependent domains. Numerical examples demonstrate the convergence properties and performance of the method.

Keegan L. A. Kirk, Tamas L. Horvath, Aycil Cesmelioglu et al.

This paper presents the first analysis of a space--time hybridizable discontinuous Galerkin method for the advection--diffusion problem on time-dependent domains. The analysis is based on non-standard local trace and inverse inequalities that are anisotropic in the spatial and time steps. We prove well-posedness of the discrete problem and provide a priori error estimates in a mesh-dependent norm. Convergence theory is validated by a numerical example solving the advection--diffusion problem on a time-dependent domain for approximations of various polynomial degree.

Esteban Henríquez, Miroslav Kuchta, Jeonghun J. Lee et al.

We propose parameter-robust preconditioners for the statically condensed linear system arising from a hybridizable discontinuous Galerkin discretization of the coupled Stokes--Darcy system. The design strategy relies on first applying the operator-preconditioning framework [Numer. Linear Algebra Appl., 18(1):1--40, 2011] to construct a preconditioner for the non-condensed discretization. This is done by proving uniform well-posedness of the scheme. Next, we prove robustness of the resulting condensed preconditioner applied to the reduced linear system using the framework we proposed in [SIAM J. Sci. Comput., 47(6):A3212--A3238, 2025]. Numerical examples demonstrate robustness of the proposed preconditioners.

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.