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, Δ²∅-bΔ∅+c∅=f₁. At each time step, the solution is represented by the sum of a volume potential and solution to Δ²∅-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. f₁ is extended to the root box of such a quad-tree by solving a biharmonic Dirichlet problem....[ Read more ]

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, Δ²∅-bΔ∅+c∅=f₁. At each time step, the solution is represented by the sum of a volume potential and solution to Δ²∅-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. f₁ 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.