TY - JOUR
T1 - A Splitting Scheme for the Numerical Solution of the KWC System
AU - R. H. W. Hoppe & J. J. Winkle
JO - Numerical Mathematics: Theory, Methods and Applications
VL - 3
SP - 661
EP - 680
PY - 2019
DA - 2019/04
SN - 12
DO - http://doi.org/10.4208/nmtma.OA-2018-0050
UR - https://global-sci.org/intro/article_detail/nmtma/13125.html
KW - Crystallization, Kobayashi-Warren-Carter system, splitting method
AB - <p style="text-align: justify;">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.</p>