Existence and stability in a virtual interpolation method of the Stokes equations

In this paper, we propose a new virtual interpolation point method to formulate the discrete Stokes equations. We form virtual staggered structure for the velocity and pressure from the actual computation node set. The virtual interpolation point method by a point collocation scheme is well suited to meshfree scheme since the approximation comes from smooth kernel and we can differentiate directly the kernels. The focus of this paper is laid on the contribution to a stable flow computation without explicit structure of staggered grid. In our method, we don't have to construct explicitly the staggered grid at all. Instead, there exists only virtual interpolation points at each computational node which play a key role in discretizing the conservative quantities of the Stokes equations. We prove the inf-sup condition for virtual interpolation point method with virtual structure of staggered grid and the existence and stability of discrete solutions.