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

Shuhao Cao, Long Chen+1 more|arXiv (Cornell University)|Mar 8, 2021

Advanced Numerical Methods in Computational MathematicsEngineering51 references33 citations

TL;DR

This paper presents a novel virtual element method (VEM) for solving two-dimensional Maxwell interface problems on unstructured, background unfitted meshes. By introducing a new H(curl)-conforming virtual space on a virtual triangulation satisfying a maximum angle condition, the method achieves first-order optimal convergence independent of mesh anisotropy near the interface, overcoming challenges from low regularity and non-conforming discretizations.

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.

Motivation & Objective

• To develop a robust numerical method for solving 2D Maxwell interface problems with discontinuous coefficients across an interface.
• To address the challenge of optimal convergence on unstructured, unfitted polygonal meshes that may become highly anisotropic near the interface.
• To construct a virtual element space that preserves H(curl)-conformity and degrees of freedom while ensuring optimal error bounds independent of element shape.
• To establish a priori error estimates that are robust with respect to mesh anisotropy and coefficient jumps.

Proposed method

• A new virtual element space is constructed as a subspace of the standard Nédélec space on a virtual triangulation of each polygonal element, satisfying a maximum angle condition.
• The virtual space shares the same degrees of freedom as the standard H(curl)-conforming VEM space, ensuring compatibility with existing VEM formulations.
• The method uses a symmetric variational formulation for the H(curl)-elliptic interface problem, with stabilization terms to handle non-conforming approximations.
• A novel analysis technique is employed, using discrete harmonic extensions defined by degrees of freedom to control boundary terms and achieve robust error estimates.
• The analysis relies on trace inequalities, Poincaré-type estimates, and approximation properties on shape-regular sub-triangles of the polygonal elements.
• The method is validated numerically on a circular interface problem with varying material parameters, using a background triangular mesh cut by the interface.

Experimental results

Research questions

• RQ1 Can a virtual element method achieve optimal convergence for Maxwell interface problems on unfitted, anisotropic polygonal meshes?
• RQ2 How can the H(curl)-conformity and degrees of freedom be preserved in a virtual space that is robust to element shape distortion?
• RQ3 What is the impact of mesh anisotropy on the convergence rate of VEM for interface problems with low-regularity solutions?
• RQ4 Can the error analysis be made independent of the aspect ratio of interface elements through a novel virtual space construction?

Key findings

• The proposed VEM achieves first-order optimal convergence in the H(curl) norm, with convergence rates of approximately 1.00 for both the L2 error and curl error, as confirmed by numerical experiments.
• The method maintains optimal convergence even when the interface elements become highly anisotropic, as demonstrated by the consistent convergence rates across all tested mesh refinements.
• The error estimates are independent of the aspect ratio of the polygonal elements, which is a key contribution for unfitted mesh methods.
• Numerical results show that the method is robust under varying material parameters (α+ = 10 or 100, β+ = 10 or 100), with convergence rates stable across all cases.
• The a priori error analysis proves that the method's convergence rate depends only on the regularity of the exact solution and the mesh size, not on element shape or anisotropy.