@article{Leng2023, 
author = {Yu Leng and Lampros Svolos and Dibyendu Adak and Ismael Boureima and Gianmarco Manzini and Hashem Mourad and Jeeyeon Plohr},
title = {A guide to the design of the virtual element methods for second- and fourth-order partial differential equations},
year = {2023},
journal = {Mathematics in Engineering},
volume = {5},
number = {6},
pages = {1-22},
keywords = {fracture mechanics, high-order phase field models, virtual element method, arbitrary-order approximations},
url = {https://www.sciopen.com/article/10.3934/mine.2023100},
doi = {10.3934/mine.2023100},
abstract = {We discuss the design and implementation details of two conforming virtual element methods for the numerical approximation of two partial differential equations that emerge in phase-field modeling of fracture propagation in elastic material. The two partial differential equations are: (i) a linear hyperbolic equation describing the momentum balance and (ii) a fourth-order elliptic equation modeling the damage of the material. Inspired by [1,2,3], we develop a new conforming VEM for the discretization of the two equations, which is implementation-friendly, i.e., different terms can be implemented by exploiting a single projection operator. We use        C    0   and        C    1   virtual elements for the second-and fourth-order partial differential equation, respectively. For both equations, we review the formulation of the virtual element approximation and discuss the details pertaining the implementation.}
}