Analysis of piecewise linear approximations to the generalized stokes problem in the velocity-stress-pressure formulation

In this paper we study continuous piecewise linear polynomial approximations to the generalized Stokes equations in the velocity-stress-pressure first-order system formulation by using a cell vertex finite volume/least-squares scheme. This method is composed of a direct cell vertex finite volume discretization step and an algebraic least-squares step, where the least-squares procedure is applied after the discretization process is accomplished. This combined approach has the advantages of both finite volume and least-squares approaches. An error estimate in the H¹ product norm for continuous piecewise linear approximating functions is derived. It is shown that, with respect to the order of approximation for H²-regular exact solutions, the method exhibits an optimal rate of convergence in the H¹ norm for all unknowns, velocity, stress, and pressure.