A Virtual Element Method for elliptic problems on trimmed background meshes

Abstract

We consider a two-dimensional piecewise $C^2$ domain that cuts through a quasi-uniform fixed polygonal background mesh, for instance made of quadrilaterals. A simple procedure based on convex hulls gives rise to a rather small number of polygonal boundary elements of various shapes, including elements with small edges and large aspect ratios; this is the computational mesh for a virtual element method (VEM), a trimmed background mesh. We classify all possible geometric configurations and study their stability and approximability properties. This entails deriving robust stabilization mechanisms and interpolation estimates for anisotropic elements and elements with small cuts, as well as a weak maximum principle for enhanced virtual elements; these contributions have intrinsic interest for VEM theory on geometric flexibility. We prove that the resulting VEM is uniformly stable in $H^1$, and also show optimal order-regularity error estimates in $H^1$ and $L^2$. Insightful numerical experiments corroborate and complement our theory. The proposed method is suitable for treating ALE formulations of problems in moving domains.