A Note on Accurate and Efficient Higher Order Galerkin Time Stepping Schemes for the Nonstationary Stokes Equations

ABSTRACT

In this note, we extend our recent work for the heat equation in [1] and describe and compare by means of numerical experiments the continuous Galerkin-Petrov (cGP) and discontinuous Galerkin (dG) time discretization applied to the nonstationary Stokes equations in the two-dimensional case. For the space discretization, we use the well-known LBB-stable quadrilateral finite element which consists of conforming biquadratic elements for the velocity and discontinuous linear elements for the pressure. We discuss implementation aspects as well as methods for solving the resulting block systems using monolithic multigrid solvers based on Vanka-type smoothers. By means of numerical experiments we compare the different time discretizations with respect to accuracy and computational costs. We show that the convergence behavior of the multigrid method is almost independent of the mesh size in space and the time step size which means (at least for these examples) that we have created an efficient solution process.

Keywords:

Discontinuous Galerkin method, Continuous Galerkin-Petrov method, Stokes equations, Multigrid solver.