An interface-fitted virtual element method for Stokes moving interface problem

This paper presents a virtual element method for a problem with a moving elastic interface in Stokes flow. On a background Cartesian mesh of the domain, each element not cut by the interface consists of eight nodes, namely, the four vertices and the mid-points of four edges. For each cut-element, we move the mid-node onto the interface location, which not only updates the element connectivity, but also allows for flexibly matching the interface as time advances. This simple and effective idea inspires us to develop an interface-fitted mesh generator. In spatial discretization, a linear virtual element approximation is developed for the Stokes problem, which delivers a velocity with local mass conservation. Then, a semi-implicit discretization is designed for the Stokes equation with immersed moving interface, where the discrete bilinear forms are concise without the use of both additional penalty terms and multipliers. Specifically, after the velocity is solved on an interface-fitted mesh, we update the Cartesian coordinates of points located on the interface and fit them with a cubic spline function to form a closed curve, which will be employed to find the intersection points of the interface with the edges of background mesh at a new time. Theoretically, we prove that the discrete scheme is unconditionally stable. Finally, the efficiency and accuracy of our method are verified by extensive numerical examples, including the optimal convergence rates in appropriate norms, the capacity to accurately track the interface evolution.