Preconditioning of a hybridized discontinuous Galerkin finite element method for the Stokes equations

TL;DR

This paper develops optimal preconditioners for a hybridized discontinuous Galerkin method solving the Stokes equations, improving computational efficiency by leveraging the pressure Schur complement's properties.

Contribution

It introduces a novel approach to preconditioning that exploits the Schur complement's spectral properties for hybridized DG discretizations of the Stokes problem.

Findings

Preconditioners show optimal performance in 2D and 3D simulations.

Spectral equivalence established between the Schur complement and simple matrices.

Numerical results confirm efficiency and robustness of the proposed preconditioners.

Abstract

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.