An Integral Equation Method for the Cahn-Hilliard Equation in the Wetting Problem
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We present an integral equation approach to solve the Cahn-Hilliard equation equipped with boundary conditions that model solid surfaces with prescribed Young's angles. Discretization of the system in time using convex splitting leads to a modified biharmonic equation at each time step. To solve it, the basic idea is to 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 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 and can achieve high order convergence with adaptive refinement.
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