THESIS
2018
xi, 67 leaves : color illustrations ; 30 cm
Abstract
The Cahn-Hilliard equation ∅
_{t} = Δ(-∈Δ∅+∅
^{3}-∅/∈) is often used in phase-field
models to describe interface problems in physics and engineering. In this
thesis, we solve the Cahn-Hilliard equation in two-dimensional domains with
solid boundary conditions. First we apply the Rothe’s method and discretize in
time using convex splitting, which leads to the modified biharmonic equation,
Δ
^{2}∅-bΔ∅+c∅=f
_{1}. At each time step, the solution is represented by the sum
of a volume potential and solution to Δ
^{2}∅-bΔ∅+c∅= 0 with proper boundary
conditions.
The volume potential is evaluated with a box based fast multipole method (Box-FMM) that assumes that the source is discretized on an adaptive quad-tree. f
_{1}
is extended to the root box of such a quad-tree by solving a biharmonic Dirichlet
problem....[
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The Cahn-Hilliard equation ∅
_{t} = Δ(-∈Δ∅+∅
^{3}-∅/∈) is often used in phase-field
models to describe interface problems in physics and engineering. In this
thesis, we solve the Cahn-Hilliard equation in two-dimensional domains with
solid boundary conditions. First we apply the Rothe’s method and discretize in
time using convex splitting, which leads to the modified biharmonic equation,
Δ
^{2}∅-bΔ∅+c∅=f
_{1}. At each time step, the solution is represented by the sum
of a volume potential and solution to Δ
^{2}∅-bΔ∅+c∅= 0 with proper boundary
conditions.
The volume potential is evaluated with a box based fast multipole method (Box-FMM) that assumes that the source is discretized on an adaptive quad-tree. f
_{1}
is extended to the root box of such a quad-tree by solving a biharmonic Dirichlet
problem. The homogeneous PDE is then cast into a second kind integral equation
(SKIE) and solved with GMRES, where the near-singular boundary integrals are
computed using quadrature by expansion (QBX) with FMM acceleration.
Our method requires solving two linear systems, both of which are well-conditioned
and only involve degrees of freedom on the domain boundary. The method is
implemented with heterogeneous parallelization using the OpenCL standard.
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