FEM∂v: FEM enhanced with boundary-layer virtual elements

Abstract

We propose F E M 𝜕 v , a boundary-enhanced discretization that retains standard finite elements on a structured background mesh and introduces lowest-order conforming virtual elements only on the polygonal cut cells adjacent to the physical boundary. The geometry is described by a level-set function or polyline and reconstructed by a two-dimensional cutting procedure that produces a controlled family of boundary-layer polygons while leaving the regular bulk mesh unchanged. For these cut virtual elements, we develop a Dirichlet-to-Neumann/Steklov-inspired stabilization that controls the non-polynomial component through computable boundary quantities and element-wise stiffness scaling. The resulting formulation combines direct treatment of the reconstructed cut geometry with the standard FEM approximation, assembly, and solver structure in the bulk. Numerical experiments for Poisson and linear-elasticity problems show the expected convergence behavior over smooth, cornered, and engineering geometries. Stabilization ablations demonstrate improved robustness for unfavorable cut configurations, and timing measurements confirm that the additional polygonal treatment remains localized to the boundary layer.