Numerical analysis of lowest-order finite volume methods for a class of Stokes variational inequality problem

Abstract

Three types of lowest-order finite volume element methods, i.e., the conforming, nonconforming and discontinuous schemes, are introduced and analysed for a variational inequality governed by the stationary Stokes equations. The variational inequality arises due to a nonlinear and nondifferentiable relationship in the slip boundary condition of friction type. This relationship cannot be well combined into a finite volume scheme by a standard procedure based on integration by parts on dual control volumes. Thereby we propose to enforce it pointwisely at cell centres in a dual mesh, which leads to some numerical integration formula for the boundary nonlinear term in the variational inequality. The resulting finite volume schemes can be seen to be equivalent to some finite element methods. We further show their solvability and stability, as well as the a priori error estimates with optimal approximation behaviours. Numerical results are reported to demonstrate the theoretical findings.