Stabilization-Free H(curl) and H(div)-Conforming Virtual Element Method

Abstract

Standard Virtual Element Method (VEM) requires stabilization terms that significantly affect the numerical computation performance. In this work, we propose a stabilization-free VEM for general order \(\mathbf{H}(\operatorname{\mathbf{curl}})\) and \(\mathbf{H}(\operatorname{div})\)-conforming spaces by constructing novel serendipity projectors and corresponding serendipity spaces with minimum number of DoFs. Our approach handles the full De Rham complex chain in \(\mathbb{R}^3\) while preserving essential properties including boundary continuity and commutativity. Since the number of DoFs are minimized, computational overhead is greatly reduced. The optimal approximation properties are rigorously proven and validated through Maxwell eigenvalue problems with numerical experiments.