Overview
This work introduces a stabilization-free Virtual Element Method (VEM) designed for general order $\mathbf{H}(\operatorname{\mathbf{curl}})}$ and $\mathbf{H}(\operatorname{\div}})$-conforming spaces. The method achieves its stabilization-free characteristic through the construction of novel serendipity projectors and corresponding serendipity spaces. A key feature of this design is the minimization of the number of Degrees of Freedom (DoFs).
Research Context
Standard Virtual Element Methods (VEM) typically rely on stabilization terms. These stabilization terms are identified as having a significant impact on the numerical computation performance of the method. The goal of this research was to address this aspect of VEM performance by developing an alternative approach.
Approach
The proposed method constructs novel serendipity projectors and corresponding serendipity spaces. These constructions are specifically designed to operate with a minimum number of DoFs. The developed approach is capable of handling the full De Rham complex chain in $\mathbb{R}^3$. Essential properties of such systems, including boundary continuity and commutativity, are preserved by this method.
Findings
- The developed VEM is stabilization-free for general order $\mathbf{H}(\operatorname{\mathbf{curl}})}$ and $\mathbf{H}(\operatorname{\div}})$-conforming spaces.
- The method utilizes novel serendipity projectors and associated serendipity spaces.
- The number of DoFs is minimized in this construction.
- The approach reduces computational overhead.
- It handles the full De Rham complex chain in $\mathbb{R}^3$.
- Essential properties such as boundary continuity and commutativity are preserved.
- Optimal approximation properties of the method have been rigorously proven.
- Numerical experiments, specifically Maxwell eigenvalue problems, were used to validate the method.
Why This Matters
The reduction in computational overhead, attributed to the minimized number of DoFs, suggests that this methodology could improve the efficiency of numerical computations in relevant applications. The preservation of essential properties like boundary continuity and commutativity ensures the method's theoretical soundness for complex systems.