Mathematical Modeling and Analysis Mimetic Finite Differences for Modeling Stokes Flow on Polygonal Meshes

Stokes flow is fluid flow where advective inertial forces are negligibly small compared to viscous forces. This is a typical situation on a microscale or when the fluid velocity is very small. Stokes flow is a good and important approximation for a number of physical problems such as sedimentation, modeling of biosuspensions, construction of efficient fibrous filters, developing energy efficient micro-fluidic devices (e.g. mixers), etc. Efficient numerical solution of Stokes flow requires unstructured meshes adapted to geometry and solution as well as accurate discretization methods capable of treating such meshes. We developed a new mimetic finite difference (MFD) method that remains accurate on general polygonal meshes that may include non-convex and degenerate elements [1]. Triangular meshes allow one to model complex geometric objects. However, compared to quadrilateral and more general polygonal meshes, the triangular meshes with the same resolution do not provide optimal cover of the space, which result in larger algebraic problems. The MFD method was designed to provide accurate approximation of differential operators on general meshes. These meshes may include degenerate elements, as in adaptive mesh refinement methods, non-convex elements, as in moving mesh methods, and even elements with curved edges near curvilinear boundaries. The incompressible Stokes equations are