Numerical verification of PolyDG algebraic solvers for the pseudo-stress Stokes problem

TL;DR

The paper tackles solver robustness for the pseudo-stress Stokes system discretized with PolyDG on polytopal meshes by examining two time-step–robust strategies: Deflated CG and collective Block-Jacobi preconditioning. It shows that both approaches yield iteration counts that remain essentially independent of the time step Δt when solving the SPD linear system A^* = M + Δt A at each implicit Euler step. Deflated CG operates by deflating the kernel of M, while collective Block-Jacobi exploits a block structure that groups all stress components per element. The results establish Δt-robustness and set the stage for extending robustness to the spatial discretization parameter h via multigrid on polytopal meshes.

Abstract

This work focuses on the development of efficient solvers for the pseudo-stress formulation of the unsteady Stokes problem, discretised by means of a discontinuous Galerkin method on polytopal grids (PolyDG). The introduction of the pseudo-stress variable is motivated by the growing interest in non-Newtonian flow models and coupled interface problems, where the stress field plays a fundamental role in the physical description. The space-time discretisation of the problem is obtained by combining the PolyDG approach in space with the implicit Euler method for time integration. The resulting linear system, characterised by a symmetric, positive, definite matrix, exhibits deteriorating convergence with standard solvers as the time step decreases. To address this issue, we investigate two tailored strategies: deflated Conjugate Gradient, which mitigates the effect of the most problematic eigenmodes, and collective Block-Jacobi, which exploits the block structure of the system matrix. Numerical experiments show that both approaches yield iteration counts effectively independent of $Δt$, ensuring robust performance with respect to the time step. Future work will focus on extending this robustness to the spatial discretisation parameter $h$ by integrating multigrid strategies with the time-robust solvers developed in this study.