Advances in Differential Equations

Finite Element Approximation of One-Sided Stefan Problems with Anisotropic, Approximately Crystalline, Gibbs--Thomson Law

John W. Barrett, Harald Garcke, and Robert Nürnberg

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Abstract

We present a finite-element approximation for the one-sided Stefan problem and the one-sided Mullins--Sekerka problem, respectively. The problems feature a fully anisotropic Gibbs--Thomson law, as well as kinetic undercooling. Our approximation, which couples a parametric approximation of the moving boundary with a finite-element approximation of the bulk quantities, can be shown to satisfy a stability bound, and it enjoys very good mesh properties, which means that no mesh smoothing is necessary in practice. In our numerical computations we concentrate on the simulation of snow crystal growth. On choosing realistic physical parameters, we are able to produce several distinctive types of snow crystal morphologies. In particular, facet breaking in approximately crystalline evolutions can be observed.

Article information

Source
Adv. Differential Equations Volume 18, Number 3/4 (2013), 383-432.

Dates
First available in Project Euclid: 5 February 2013

Permanent link to this document
https://projecteuclid.org/euclid.ade/1360073021

Mathematical Reviews number (MathSciNet)
MR3060200

Zentralblatt MATH identifier
1271.80005

Subjects
Primary: 80A22: Stefan problems, phase changes, etc. [See also 74Nxx] 74N05: Crystals 65M60: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 35R37: Moving boundary problems 65M12: Stability and convergence of numerical methods 80M10: Finite element methods

Citation

Barrett, John W.; Garcke, Harald; Nürnberg, Robert. Finite Element Approximation of One-Sided Stefan Problems with Anisotropic, Approximately Crystalline, Gibbs--Thomson Law. Adv. Differential Equations 18 (2013), no. 3/4, 383--432. https://projecteuclid.org/euclid.ade/1360073021.


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