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.