BDDC preconditioners for divergence free virtual element discretizations of the Stokes equations

TL;DR

This paper extends BDDC preconditioners to divergence-free virtual element discretizations of the Stokes equations, demonstrating theoretical convergence, scalability, and robustness on polygonal meshes with numerical validation.

Contribution

It introduces a BDDC preconditioner tailored for VEM discretizations of the Stokes problem, including a convergence analysis and practical implementation insights.

Findings

Preconditioned system is symmetric and positive definite under certain conditions.

Numerical experiments confirm scalability and quasi-optimality.

Method is robust to polygonal mesh shape variations.

Abstract

The Virtual Element Method (VEM) is a new family of numerical methods for the approximation of partial differential equations, where the geometry of the polytopal mesh elements can be very general. The aim of this article is to extend the balancing domain decomposition by constraints (BDDC) preconditioner to the solution of the saddle-point linear system arising from a VEM discretization of the two-dimensional Stokes equations. Under suitable hypotesis on the choice of the primal unknowns, the preconditioned linear system results symmetric and positive definite, thus the preconditioned conjugate gradient method can be used for its solution. We provide a theoretical convergence analysis estimating the condition number of the preconditioned linear system. Several numerical experiments validate the theoretical estimates, showing the scalability and quasi-optimality of the method proposed. Moreover, the solver exhibits a robust behavior with respect to the shape of the polygonal mesh elements. We also show that a faster convergence could be achieved with an easy to implement coarse space, slightly larger than the minimal one covered by the theory.