TY - JOUR
T1 - A Time Splitting Space Spectral Element Method for the Cahn-Hilliard Equation
AU - Lizhen Chen & Chuanju Xu
JO - East Asian Journal on Applied Mathematics
VL - 4
SP - 333
EP - 351
PY - 2018
DA - 2018/02
SN - 3
DO - http://doi.org/10.4208/eajam.150713.181113a
UR - https://global-sci.org/intro/article_detail/eajam/10861.html
KW - Cahn-Hilliard, time splitting schemes, spectral methods, error analysis.
AB - <p style="text-align: justify;">We propose and analyse a class of fully discrete schemes for the Cahn-Hilliard
equation with Neumann boundary conditions. The schemes combine large-time step
splitting methods in time and spectral element methods in space. We are particularly
interested in analysing a class of methods that split the original Cahn-Hilliard equation
into lower order equations. These lower order equations are simpler and less
computationally expensive to treat. For the first-order splitting scheme, the stability
and convergence properties are investigated based on an energy method. It is proven
that both semi-discrete and fully discrete solutions satisfy the energy dissipation and
mass conservation properties hidden in the associated continuous problem. A rigorous
error estimate, together with numerical confirmation, is provided. Although not
yet rigorously proven, higher-order schemes are also constructed and tested by a series
of numerical examples. Finally, the proposed schemes are applied to the phase field
simulation in a complex domain, and some interesting simulation results are obtained.</p>