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Virtual Element Method

The virtual element approximations to the solutions of the two sample problems

Introduction

The Virtual Element Method (VEM) is a technique for numerically approximating partial differential equations using generalised polygonal or polytopedral meshes. It is an extension of Mimetic Finite Differences (MFD) and Finite Element Methods (FEM).

History

The method was first introduced in 2013 by L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo.

Technical descussion

The structure of virtual element methods

Similar to the FEM it is characterized by a variational formulation

The continuous problem

$$\left\{{\begin{aligned}-\Delta u&=f\quad {\text{in }}\Omega ,\\u&=0\quad {\text{on }}\partial \Omega \end{aligned}}\right.$$

where $f\in L^{2}(\Omega )$ and $\Omega \subseteq \mathbb {R} ^{n}$.

Discretization

The domain $\Omega$ can be discretized into a finite number of partitions $\left\{{\mathcal {T}}_{h}\right\}_{h}$.

Virtual Element Space

Formally the virtual element space is defined as: $$V_{h}:=\{v_{h}\in H_{0}^{1}(\Omega ):v_{h}|_{E}\in V_{h}^{E}\,\forall \,E\in {\mathcal {T}}_{h}\}$$ where:

• $V_{h}$ is a subspace of $H_{0}^{1}(\Omega )$, the Sobolev space of functions that are square-integrable with square-integrable first derivatives, and that vanish on the boundary of $\Omega$.
• $V_{h}^{E}$ represents the local virtual element space associated with each element $E$ in the partition ${\mathcal {T}}_{h}$.

The virtual element space allows for flexibility in the choice of element shapes, accommodating general polygonal and polyhedral meshes, which is a significant advantage over traditional finite element methods.

Ritz projection

The Ritz projection is defined as the operator $\Pi ^{E}:V_{h}^{E}\rightarrow {\mathcal {P}}_{E}$, which satisfies the following two properties:

1. $(\nabla (\Pi ^{E}v_{h}-v_{h}),\nabla p)_{0,E}=0\quad \forall p\in {\mathcal {P}}_{E}$
2. ${\overline {\Pi ^{E}v_{h}}}={\overline {v_{h}}}$

Applications

VEM is applied in various fields including structural mechanics, fluid dynamics, and electromagnetics. For example, it has been used to model complex geometries in engineering problems.

Advantages and Limitations

VEM offers advantages such as flexibility in mesh generation and improved accuracy for certain problems. However, it may require more computational resources compared to traditional methods.

See also

• Polytope

Further Reading

• Beirão da Veiga, L., Brezzi, F., Marini, L. D., & Russo, A. (2014). The Hitchhiker’s Guide to the Virtual Element Method. Springer.
• Sutton, O. J. (2017). "The virtual element method in 50 lines of MATLAB". Numerical Algorithms, 75(4), 1141-1159. https://doi.org/10.1007/s11075-016-0235-3