  We present an integral equation approach to solving the Cahn-Hilliard
equation equipped with boundary conditions that model solid surfaces with
prescribed Young's angles. The discretization of the system in time using
convex splitting leads to a modified biharmonic equation at each time step. To
solve it, we split the solution into a volume potential computed with free
space kernels, plus the solution to a second kind integral equation (SKIE). The
volume potential is evaluated with the help of a box-based volume-FMM method.
For non-box domains, source density is extended by solving a biharmonic
Dirichlet problem. The near-singular boundary integrals are computed using
quadrature by expansion (QBX) with FMM acceleration. Our method has linear
complexity in the number of surface/volume degrees of freedom and can achieve
high order convergence with adaptive refinement to manage error from function
extension.
