Analysis of Stabilized Finite Volume Method for the Transient Stokes Equations

Abstract

This paper is concerned with the development and study of a stabilized finite volume method for the transient Stokes problem in two and three dimensions. The stabilization is based on two local Gauss integrals and is parameter-free. The analysis is based on a relationship between this new finite volume method and a stabilized finite element method using the lowest equal-order pair (i.e., the $P_1$-$P_1$ pair). An error estimate of optimal order in the $H^1$-norm for velocity and an estimate in the $L^2$-norm for pressure are obtained. An optimal error estimate in the $L^2$-norm for the velocity is derived under an additional assumption on the body force.