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A \(C^1\)-Conforming Arbitrary-Order Two-Dimensional Virtual Element Method for the Fourth-Order Phase-Field Equation

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Abstract

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.

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Data Availability

No new data were specifically generated for this work. The meshes utilized for this study and reported in the final appendix are the property of Los Alamos National Laboratory and can be made available on reasonable request.

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Acknowledgements

The authors gratefully acknowledge the support of the Laboratory Directed Research and Development (LDRD) program of Los Alamos National Laboratory under project number 20220129ER. Los Alamos National Laboratory is operated by Triad National Security, LLC, for the National Nuclear Security Administration of U.S. Department of Energy (Contract No. 89233218CNA000001). This work is registered as the Los Alamos Technical Report LA-UR-21-31995.

Funding

Laboratory Directed Research and Development (LDRD) program operated at Los Alamos National Laboratory, Grant N. 20220129ER. All the Authors have no financial interests

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Authors and Affiliations

  1. GIMNAP, Departamento de Matemática, Universidad del Bío-Bío, Concepción, Chile

    Dibyendu Adak

  2. Applied Mathematics and Plasma Physics, T-5, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

    Gianmarco Manzini

  3. Fluid Dynamics and Solid Mechanics, T-3, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, USA

    Hashem M. Mourad & Lampros Svolos

  4. Materials and Physical Data, XCP-5, Computational Physics Division, Los Alamos National Laboratory, Los Alamos, NM, USA

    JeeYeon N. Plohr

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  1. Dibyendu Adak
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Correspondence to Gianmarco Manzini.

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Appendix A: Mesh Data

Appendix A: Mesh Data

See Tables 1, 2, 3, 4.

Table 1 Mesh data and number of degrees of freedom for the VEM with polynomial degree \(k=2,3,4\) for the family of smoothly remapped quadrilateral meshes
Full size table
Table 2 Mesh data and number of degrees of freedom for the VEM with polynomial degree \(k=2,3,4\) for the family of randomized quadrilateral meshes
Full size table
Table 3 Mesh data and number of degrees of freedom for the VEM with polynomial degree \(k=2,3,4\) for the family of smoothly remapped hexagons
Full size table
Table 4 Mesh data and number of degrees of freedom for the VEM with polynomial degree \(k=2,3,4\) for the family of nonconvex octagons
Full size table

For completeness, we report the data corresponding to the four mesh families that we used in the calculation of Sectionsec6:numerical:results. The four tables report the following data:

  • \(n\): refinement level;

  • \(N_{P}\), \(N_{F}\), \(N_{V}\): number of elements, edges, and vertices;

  • \(h\): mesh size parameter;

  • \(\#\text {dofs}_{k=\ell }\): total number of degrees of freedom for the VEM with polynomial degree \(\ell =2,3,4\).

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Adak, D., Manzini, G., Mourad, H.M. et al. A \(C^1\)-Conforming Arbitrary-Order Two-Dimensional Virtual Element Method for the Fourth-Order Phase-Field Equation. J Sci Comput 98, 38 (2024). https://doi.org/10.1007/s10915-023-02409-w

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