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

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 them 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 of Cao, Chen, and Guo [Math. Models Methods Appl. Sci. 33 (2023), pp. 455–503] and Chen, Wei, and Wen [J. Comput. Phys. 334 (2017), pp. 327–348] that employ 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 whose convex hulls contain non-shrinking inscribed balls, with a non-shrinking volume condition; (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 cases widely appear in generating fitted meshes for interface and fracture problems, while the third one is used on unfitted meshes for interface problems. The present research allows the VEMs to go beyond star-convex and non-shrinking element shapes and also beyond elements containing only one material. In addition, this work generalizes the maximum angle condition from simplicial meshes to polyhedral meshes.