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

Yu Leng; Lampros Svolos; Dibyendu Adak; Ismael Boureima; Gianmarco Manzini; Hashem Mourad; Jeeyeon Plohr; Yu Leng; Lampros Svolos; Dibyendu Adak; Ismael Boureima; Gianmarco Manzini; Hashem Mourad; Jeeyeon Plohr

doi:10.3934/mine.2023100

Mathematics in Engineering

2023, Volume 5, Issue 6: 1-22. doi: 10.3934/mine.2023100

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

1.
T-3, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
2.
T-5, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA
3.
XCP-5, Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USA

Received: 13 August 2023 Revised: 21 October 2023 Accepted: 24 October 2023 Published: 09 November 2023

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.
- fracture mechanics,
- high-order phase field models,
- virtual element method,
- arbitrary-order approximations
Citation: Yu Leng, Lampros Svolos, Dibyendu Adak, Ismael Boureima, Gianmarco Manzini, Hashem Mourad, Jeeyeon Plohr. A guide to the design of the virtual element methods for second- and fourth-order partial differential equations[J]. Mathematics in Engineering, 2023, 5(6): 1-22. doi: 10.3934/mine.2023100

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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.

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