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：本文针对最低阶的面元和边元虚拟元方法，通过在多边形网格上推导离散正则分解，并将Hiptmair-Xu预条件子推广到虚拟元框架中，提出了节点辅助空间预条件子。该预条件子包含在节点虚拟元空间上求解一系列椭圆问题，并结合适当的平滑步骤。在网格正则性假设下，证明预条件系统的谱条件数有界且与网格尺寸无关，并通过一系列多边形网格上的数值实验验证了这一结论。此外，数值结果表明该预条件子对包含高长宽比单元的网格具有鲁棒性。