A discontinuous viscosity coefficient makes the jump conditions of the velocity and normal stress coupled together, which brings great challenges to some commonly used numerical methods to obtain accurate solutions. To overcome the difficulties, a kernel free boundary integral (KFBI) method combined with a modified marker-and-cell (MAC) scheme is developed to solve the two-phase Stokes problems with discontinuous viscosity. The main idea is to reformulate the two-phase Stokes problem into a single-fluid Stokes problem by using boundary integral equations and then evaluate the boundary integrals indirectly through a Cartesian grid-based method. Since the jump conditions of the single-fluid Stokes problems can be easily decoupled, the modified MAC scheme is adopted here and the existing fast solver can be applicable for the resulting linear saddle system. The computed numerical solutions are second order accurate in discrete $\ell^2$-norm for velocity and pressure as well as the gradient of velocity, and also second order accurate in maximum norm for both velocity and its gradient, even in the case of high contrast viscosity coefficient, which is demonstrated in numerical tests.

翻译：间断粘性系数使得速度和法向应力的跳跃条件相互耦合，这给许多常用数值方法获取精确解带来了巨大挑战。为克服这一困难，本文发展了一种结合改进型标记-单元（MAC）格式的无核边界积分（KFBI）方法，用于求解具有间断粘性的两相Stokes问题。核心思想是通过边界积分方程将两相Stokes问题重新表述为单流体Stokes问题，并利用基于笛卡尔网格的方法间接计算边界积分。由于单流体Stokes问题的跳跃条件易于解耦，本文采用改进型MAC格式，且现有快速求解器可应用于由此产生的线性鞍点系统。数值试验表明，即使在粘性系数高对比度的情况下，所获数值解在离散$\ell^2$范数下速度、压力及速度梯度均具有二阶精度，同时在最大范数下速度及其梯度也达到二阶精度。