On the maximum angle conditions for polyhedra with virtual element methods

TL;DR

This paper extends maximum angle conditions to polyhedral meshes for virtual element methods, providing an anisotropic analysis framework that ensures shape-independent error estimates and broadens VEM applications.

Contribution

It develops a novel anisotropic analysis framework for VEMs that accommodates complex polyhedral shapes and material discontinuities, improving theoretical understanding and applicability.

Findings

Error estimates are shape-independent.

Framework applies to non-star convex and arbitrarily cut elements.

Broadens VEM applicability to complex polyhedral meshes.

Abstract

Finite element methods are well-known to admit robust optimal convergence on simplicial meshes satisfying the maximum angle conditions. But how to generalize this condition to polyhedra is unknown in the literature. In this work, we argue that this generation is possible for virtual element methods (VEMs). In particular, we develop an anisotropic analysis framework for VEMs where the virtual spaces and projection spaces remain abstract and can be problem-adapted, carrying forward the ``virtual'' spirit of VEMs. Three anisotropic cases will be analyzed under this framework: (1) elements only contain non-shrinking inscribed balls but 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 error estimates are guaranteed to be independent of polyhedral element shapes. The present work largely improves the current theoretical results in the literature and also broadens the scope of the application of VEMs.