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 β2-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