Preconditioning of a hybridizable discontinuous Galerkin method for the coupled Stokes--Darcy system

TL;DR

This paper develops parameter-robust preconditioners for a hybridizable discontinuous Galerkin method applied to the coupled Stokes--Darcy system, ensuring efficiency and stability across various parameters.

Contribution

It introduces a novel preconditioning strategy based on operator-preconditioning and proves its robustness for the coupled Stokes--Darcy system discretized by HDG methods.

Findings

Numerical examples confirm the robustness of the proposed preconditioners.

The preconditioners are effective for the reduced linear system.

The approach ensures parameter-robustness in the discretization.

Abstract

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.