Volume 12, Issue 3
A Splitting Scheme for the Numerical Solution of the KWC System

R. H. W. Hoppe and J. J. Winkle

10.4208/nmtma.OA-2018-0050

Numer. Math. Theor. Meth. Appl., 12 (2019), pp. 661-680.

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  • Abstract

We consider a splitting method for the numerical solution of the regularized Kobayashi-Warren-Carter (KWC) system which describes the growth of single crystal particles of different orientations in two spatial dimensions. The KWC model is a system of two nonlinear parabolic PDEs representing gradient flows associated with a free energy in two variables. Based on an implicit time discretization by the backward Euler method, we suggest a splitting method and prove the existence as well as the energy stability of a solution. The discretization in space is taken care of by Lagrangian finite elements with respect to a geometrically conforming, shape regular, simplicial triangulation of the computational domain and requires the successive solution of two individual discrete elliptic problems. Viewing the time as a parameter, the fully discrete equations represent a parameter dependent nonlinear system which is solved by a predictor corrector continuation strategy with an adaptive choice of the time step size. Numerical results illustrate the performance of the splitting method.

  • History

Published online: 2019-04

  • AMS Subject Headings

65M12,35K59,74N05

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