Virtual element methods based on boundary triangulation:fitted and unfitted meshes

TL;DR

This paper provides a rigorous analysis of optimal convergence for virtual element methods on complex 3D polyhedral meshes, including anisotropic and unfitted meshes, extending the maximum angle condition to polyhedral cases.

Contribution

It establishes theoretical optimal convergence results for VEMs on various complex polyhedral meshes, filling a significant gap in the analysis of these methods.

Findings

Proves optimal convergence for non-star convex polyhedral meshes.

Shows robustness of VEMs on arbitrarily cut meshes from Cartesian grids.

Extends maximum angle condition to polyhedral meshes.

Abstract

One remarkable feature of virtual element methods (VEMs) is their great flexibility and robustness when used on almost arbitrary polytopal meshes. This very feature makes it widely used in both fitted and unfitted mesh methods. Despite extensive numerical studies, a rigorous analysis of robust optimal convergence has remained open for highly anisotropic 3D polyhedral meshes. In this work, we consider the VEMs in \cite{2023CaoChenGuo,2017ChenWeiWen} that introduce a boundary triangulation satisfying the maximum angle condition. We close this theoretical gap regarding optimal convergence on polyhedral meshes in the lowest-order case for the following three types of meshes: (1) elements only contain non-shrinking inscribed balls but \textit{are not necessarily star convex} to those balls; (2) elements are cut arbitrarily from a background Cartesian mesh, which can extremely shrink; (3) elements contain different materials on which the virtual spaces involve discontinuous coefficients. The first two widely appear in generating fitted meshes for interface and fracture problems, while the third one is used for unfitted mesh on interface problems. In addition, the present work also generalizes the maximum angle condition from simplicial meshes to polyhedral meshes.