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

TL;DR

This paper develops adaptive BDDC preconditioners with frugal coarse spaces for virtual element discretizations of Stokes problems with heterogeneous viscosity, improving robustness and efficiency.

Contribution

It introduces new adaptive techniques to enrich BDDC coarse spaces specifically for VEM discretizations of Stokes problems with variable viscosity, including a computationally cheaper heuristic.

Findings

The heuristic approach combined with deluxe scaling performs well.

Numerical results demonstrate improved preconditioning effectiveness.

The methods are suitable for complex polygonal and polyhedral grids.

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.