We present a two-dimensional conforming virtual element method for the fourth-order phase-field equation. Our proposed numerical approach to the solution of this high-order phase-field (HOPF) equation relies on the design of an arbitrary-order accurate, virtual element space with $C^1$ global regularity. Such regularity is guaranteed by taking the values of the virtual element functions and their full gradient at the mesh vertices as degrees of freedom. Attaining high-order accuracy requires also edge polynomial moments of the trace of the virtual element functions and their normal derivatives. In this work, we detail the scheme construction, and prove its convergence by deriving error estimates in different norms. A set of representative test cases allows us to assess the behavior of the method.
Adak, Dibyendu, et al. "A $C^1$-Conforming Arbitrary-Order Two-Dimensional Virtual Element Method for the Fourth-Order Phase-Field Equation." Journal of Scientific Computing, vol. 98, no. 2, Jan. 2024. https://doi.org/10.1007/s10915-023-02409-w
Adak, Dibyendu, Manzini, Gianmarco, Mourad, Hashem Mohamed, et al., "A $C^1$-Conforming Arbitrary-Order Two-Dimensional Virtual Element Method for the Fourth-Order Phase-Field Equation," Journal of Scientific Computing 98, no. 2 (2024), https://doi.org/10.1007/s10915-023-02409-w
@article{osti_2574137,
author = {Adak, Dibyendu and Manzini, Gianmarco and Mourad, Hashem Mohamed and Plohr, JeeYeon Nam and Svolos, Lampros},
title = {A $C^1$-Conforming Arbitrary-Order Two-Dimensional Virtual Element Method for the Fourth-Order Phase-Field Equation},
annote = {We present a two-dimensional conforming virtual element method for the fourth-order phase-field equation. Our proposed numerical approach to the solution of this high-order phase-field (HOPF) equation relies on the design of an arbitrary-order accurate, virtual element space with $C^1$ global regularity. Such regularity is guaranteed by taking the values of the virtual element functions and their full gradient at the mesh vertices as degrees of freedom. Attaining high-order accuracy requires also edge polynomial moments of the trace of the virtual element functions and their normal derivatives. In this work, we detail the scheme construction, and prove its convergence by deriving error estimates in different norms. A set of representative test cases allows us to assess the behavior of the method.},
doi = {10.1007/s10915-023-02409-w},
url = {https://www.osti.gov/biblio/2574137},
journal = {Journal of Scientific Computing},
issn = {ISSN 0885-7474},
number = {2},
volume = {98},
place = {United States},
publisher = {Springer},
year = {2024},
month = {01}}