An augmented approach for Stokes equations with discontinuous viscosity and singular forces

For Stokes equations with discontinuous viscosity across an arbitrary interface or/and singular forces along the interface, the pressure is known to be discontinuous and the velocity is known to be non-smooth. It has been shown that these discontinuities are coupled together which makes it difficult to obtain accurate numerical solutions. In this paper, a second order accurate numerical method that decouples the jump conditions of the fluid variables through two augmented variables has been developed. The GMRES iterative method is used to solve the Schur complement system for the augmented variables which are only defined on the interface. The augmented approach also rescales the Stokes equations in such a way that fast Poisson solvers can be used in each iteration. Numerical examples against exact solutions show that the new method has average second order accuracy in the infinity norm, and the number of GMRES iterations is independent of mesh sizes. An example of a moving interface problem is also presented.