A NEW FINITE VOLUME METHOD FOR THE STOKES PROBLEMS

In scientific computing for science and engineering problems, finite volume methods are widely used and appreciated by users due to their local conservative properties for quantities which are of practical interest (e.g., mass or energy). Among many references, we would like to cite some which addresses theoretical issues such as stability and convergence [5, 6, 10, 11, 15, 16, 20, 21, 22, 8, 9, 10, 28, 29]. The goal of this paper is to investigate a finite volume method for the Stokes equations by using the well-known BDM elements [3] originally designed for solving second order elliptic problems. We intend to demonstrate how the BDM element can be employed in constructing finite volume methods for the model Stokes equations. The idea to be presented in the paper can be extended to problems of Stokes and Navier-Stokes type without any difficulty. Mass conservation is a property that numerical schemes should sustain in computational fluid dynamics. This property is often characterized as an incompressibility constraint in the modeling equations. To sustain the mass conservation property for the Stokes equations, several finite element schemes have been developed to generate locally divergence-free solutions [12, 23]. In particular, a recent approach by using H(div) conforming finite elements has been proposed and studied for a numerical approximation of incompressible fluid flow problems [13, 25, 26]. The main