A Virtual Finite Element Method for Two Dimensional Maxwell Interface Problems with a Background Unfitted Mesh

TL;DR

This paper introduces a virtual element method for 2D Maxwell interface problems on polygonal meshes cut by interfaces, achieving optimal convergence and error bounds independent of mesh anisotropy.

Contribution

A novel virtual space on a virtual triangulation is developed, enabling optimal error bounds for Maxwell interface problems on unfitted polygonal meshes.

Findings

Achieves first-order optimal convergence.

Error bounds are independent of mesh anisotropy.

Applicable to special polygonal meshes cut by interfaces.

Abstract

A virtual element method (VEM) with the first order optimal convergence order is developed for solving two-dimensional Maxwell interface problems on a special class of polygonal meshes that are cut by the interface from a background unfitted mesh. A novel virtual space is introduced on a virtual triangulation of the polygonal mesh satisfying a maximum angle condition, which shares exactly the same degrees of freedom as the usual H(curl)-conforming virtual space. This new virtual space serves as the key to prove that the optimal error bounds of the VEM are independent of high aspect ratio of the possible anisotropic polygonal mesh near the interface.