Adaptive and frugal BDDC coarse spaces for virtual element discretizations of a Stokes problem with heterogeneous viscosity

TL;DR

This work tackles ill-conditioning in virtual element discretizations of a Stokes problem with highly heterogeneous viscosity by developing adaptive and frugal BDDC coarse spaces. It extends two adaptive coarse-space strategies and a computationally cheap frugal heuristic to the Stokes-VEM setting, incorporating edge-based eigenproblems and a generalized basis transformation. The preconditioner with deluxe scaling demonstrates strong robustness and can substantially reduce primal constraints, making the frugal approach competitive in practice. Numerical tests on realistic viscosity distributions confirm the proposed methods' effectiveness across different mesh types and partitionings, highlighting practical impact for scalable solvers in complex fluid problems.

Abstract

The virtual element method (VEM) is a family of numerical methods to discretize partial differential equations on general polygonal or polyhedral computational grids. However, the resulting linear systems are often ill-conditioned and robust preconditioning techniques are necessary for an iterative solution. Here, a balancing domain decomposition by constraints (BDDC) preconditioner is considered. Techniques to enrich the coarse space of BDDC applied to a Stokes problem with heterogeneous viscosity are proposed. In this framework a comparison between two adaptive techniques and a computationally cheaper heuristic approach is carried out. Numerical results computed on a physically realistic model show that the latter approach in combination with the deluxe scaling is a promising alternative.

Figures (1)

• Figure 1: METIS decomposition of a RND mesh. $\nu = 1e3$ on yellow elements, $\nu = 1e-3$ on blue ones.