Analysis of the PSPG method for the transient Stokes’ problem

We propose a new analysis for the PSPG method applied to the transient Stokes’ problem. Stability and convergence are obtained under different conditions on the discretization parameters depending on the approximation used in space. For the pressure we prove optimal stability and convergence only in the case of piecewise affine approximation under the standard condition on the time-step. Finally we show that the stability problems of the PSPG-method can be circumvented using an appropriate discrete reconstruction of the Laplace operator.

► New stability estimates for pressure stabilized Petrov-Galerkin methods for the Stokes’ problem.
► Optimal convergence estimates for the velocity approximation in natural norms.
► Stability and convergence of the pressure for piecewise affine approximation spaces in the natural norm.
► Equivalence of some known stabilized methods for Stokes problem.
► Numerical example showing the small time-step velocity instability for high order finite element approximations.