Error analysis for the pseudostress formulation of unsteady Stokes problem

In this paper, we are concerned with error analysis of the semi-discrete and fully discrete approximations to the pseudostress-velocity formulation of the unsteady Stokes problem. The pseudostress-velocity formulation of the Stokes problem allows a Raviart-Thomas mixed finite element. For the semi-discrete approximation, we prove that solution operators of homogeneous Stokes equations have the so-called parabolic smoothing property. For the fully discrete case, backward Euler and Crank-Nicolson schemes in time are considered. We present how to find the initial value of the pseudostress variable which is not given as initial data in Crank-Nicolson algorithm. Matrix equations are derived to show that backward Euler and Crank-Nicolson schemes corresponding to the pseudostress-velocity formulation are unconditionally stable. Finally, numerical examples are presented to test the performance of the algorithm and validity of the theory developed.