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, the 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 in space with adaptive refinement to manage error from function extension.

我们提出了一种积分方程方法，可以解决带有边界条件的Cahn-Hilliard方程，该边界条件以规定的杨氏角对实体表面进行建模。使用凸分裂在时间上对系统进行离散化会在每个时间步长处生成一个经过修改的双谐波方程。为了解决该问题，我们将解决方案拆分为使用自由空间核计算的体势，再加上第二类积分方程（SKIE）的解决方案。借助基于盒的体积FMM方法评估体积潜力。对于非盒域，通过解决双调和Dirichlet问题来扩展源密度。使用具有FMM加速度的正交展开（QBX）计算近似奇异边界积分。