Nodal auxiliary space preconditioners for mixed virtual element methods

TL;DR

This paper introduces a new preconditioning technique for virtual element methods that ensures efficient and robust solutions on complex polygonal meshes, with proven mesh-independent performance.

Contribution

It extends the Hiptmair-Xu preconditioner to virtual element frameworks using discrete regular decompositions, improving solver robustness and efficiency.

Findings

Preconditioner achieves mesh-independent spectral bounds.

Numerical tests confirm robustness on high aspect ratio meshes.

Method effectively solves elliptic problems on polytopal grids.

Abstract

We propose nodal auxiliary space preconditioners for facet and edge virtual elements of lowest order by deriving discrete regular decompositions on polytopal grids and generalizing the Hiptmair-Xu preconditioner to the virtual element framework. The preconditioner consists of solving a sequence of elliptic problems on the nodal virtual element space, combined with appropriate smoother steps. Under assumed regularity of the mesh, the preconditioned system is proven to have bounded spectral condition number independent of the mesh size and this is verified by numerical experiments on a sequence of polygonal meshes. Moreover, we observe numerically that the preconditioner is robust on meshes containing elements with high aspect ratios.