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

摘要

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