Grad-div Stabilized Finite Element Schemes for the Fluid-Fluid Interaction Model

In this work, two fully discrete grad-div stabilized finite element schemesfor the fluid-fluid interaction model are considered. The first scheme is standard grad-div stabilized scheme, and the other one is modular grad-div stabilized scheme whichadds to Euler backward scheme an update step and does not increase computationaltime for increasing stabilized parameters. Moreover, stability and error estimates ofthese schemes are given. Finally, computational tests are provided to verify both thenumerical theory and efficiency of the presented schemes.     

A Simple 3D Immersed Interface Method for Stokes Flow with Singular Forces on Staggered Grids

In this paper, a fairly simple 3D immersed interface method based on theCG-Uzawa type method and the level set representation of the interface is employedfor solving three-dimensional Stokes flow with singular forces along the interface. Themethod is to apply the Taylor's expansions only along the normal direction and incorporate the jump conditions up to the second normal derivatives into the finite difference schemes. A second order geometric iteration algorithm is employed for computing orthogonal projections on the surface with third-order accuracy. The Stokesequations are discretized involving the correction terms on staggered grids and thensolved by the conjugate gradient Uzawa type method. The major advantages of thepresent method are the special simplicity, the ability in handling the Dirichlet boundary conditions, and no need of the pressure boundary condition. The method canalso preserve the volume conservation and the discrete divergence free condition verywell. The numerical results show that the proposed method is second order accurateand efficient.