Fast integral equation methods for Cahn-Hilliard equation

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, 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.

Date of Award	2018
Original language	English
Awarding Institution	The Hong Kong University of Science and Technology