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.

翻译：通过在多边形网格上推导离散正则分解并将Hiptmair-Xu预条件子推广至虚拟元框架，我们为最低阶的面与边虚拟元构造了节点辅助空间预条件子。该预条件子由节点虚拟元空间上的一系列椭圆问题求解与适当的平滑步骤组合而成。在假定网格正则的条件下，我们证明了预条件系统的谱条件数有界且与网格尺寸无关，这一结论通过多边形网格序列上的数值实验得到验证。此外，数值实验表明该预条件子对包含高纵横比单元的网格具有鲁棒性。