Hybrid discontinuous Galerkin discretisation and domain decomposition preconditioners for the Stokes problem

TL;DR

This paper introduces a hybrid discontinuous Galerkin method combined with a novel domain decomposition preconditioner to efficiently solve the Stokes problem, demonstrating improved performance through detailed analysis and numerical experiments.

Contribution

It presents a new coupling of hybrid discontinuous Galerkin discretisation with an innovative domain decomposition preconditioner for the Stokes system, with thorough analysis and validation.

Findings

The hybrid discontinuous Galerkin method effectively handles non standard interface conditions.

The new preconditioners outperform classical choices in numerical experiments.

Numerical evidence supports the theoretical advantages of the proposed approach.

Abstract

Solving the Stokes equation by an optimal domain decomposition method derived algebraically involves the use of non standard interface conditions whose discretisation is not trivial. For this reason the use of approximation methods such as hybrid discontinuous Galerkin appears as an appropriate strategy: on the one hand they provide the best compromise in terms of the number of degrees of freedom in between standard continuous and discontinuous Galerkin methods, and on the other hand the degrees of freedom used in the non standard interface conditions are naturally defined at the boundary between elements. In this paper we introduce the coupling between a well chosen discretisation method (hybrid discontinuous Galerkin) and a novel and efficient domain decomposition method to solve the Stokes system. We present the detailed analysis of the hybrid discontinuous Galerkin method for the Stokes problem with non standard boundary conditions. This analysis is supported by numerical evidence. In addition, the advantage of the new preconditioners over more classical choices is also supported by numerical experiments.