A posteriori error analysis and mesh adaptivity for a virtual element method solving the Stokes equations

Abstract

We investigate an adaptive mesh strategy for the conforming virtual element method (VEM) of the Stokes equations proposed in Manzini and Mazzia (2022). The VEM generalizes the finite element approach to polygonal and polyehedral meshes in the framework of Galerkin approximation. The scheme of Manzini and Mazzia (2022) is inf-sup stable, converges optimally in the L2 and energy norm for all polynomial orders k≥1, and the Stokes velocity is weakly divergence-free at the machine precision level. Our adaptive mesh strategy is based on a suitable residual-based a posteriori indicator. A posteriori analysis shows that such an indicator is theoretically efficient and reliable. Our numerical experiments show that it can be an efficient tool for solving scientific and engineering problems by applying it to a set of representative situations, including the case of a weakly singular solution as that of the “L-shape” domain.