A guide to the design of the virtual element methods for second- and fourth-order partial differential equations

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.