# Continuous vs discontinuous space-time FEM

What are some reasons for approximating a (e.g. parabolic) PDE using the space-time method, with continuous finite elements in time, vs discontinuous finite elements in time?

Are there e.g. significant differences in accuracy, or stability/spurious oscillations...?

I'm particularly interested in a comparison taking into account:

• solving speed
• accuracy
• extensibility: to nonlinearities, inhomogeneous and time-dependent boundary conditions, moving meshes...

I couldn't find any exhaustive references out there.

More concretely, it can be shown that discontinuous Galerkin (dG(r)) schemes lead to strongly A-stable time stepping schemes and continuous Galerkin (cG(r)) schemes are A-stable time stepping schemes, see Besier's dissertation. A comparison of continuous and discontinuous time discretizations can be found in von Danwitz et al. Using full space-time FEM can easily also be applied to nonlinear PDEs. The same goes for tensor-product space-time FEM, where you have a seperate finite element basis for time and for space. The only difficulty there is that the system matrix can no longer be expressed as a kronecker product of temporal and spatial matrices, but you can still apply Newton's method as usual and we applied tensor-product space-time FEM to the nonlinear Navier-Stokes equations (journal paper). (The preprint of this paper is (Navier-Stokes preprint).) My colleague Jan Philipp Thiele simplified the code from the paper and started working on a user-friendly tensor-product space-time FEM library based on deal.II, which you can find in this GitHub repository. There you can also find a space-time FEM code for the Navier-Stokes equations. Space-time FEM can also be applied to moving domains. For this I would recommend works by Max von Danwitz, Norbert Hosters, Marek Behr or papers from papers from Christoph Lehrenfeld's group who implement this in their ngsxfem library. Concerning speed, you obviously pay for higher accuracy by solving larger linear systems, e.g. going from dG(0) to dG(1) you double the number of space-time DoFs. Nevertheless, sometimes it is worth it since for fluid problems dG(0) damps out the solution flow too much and you can only resolve the flow well by using dG(1) or higher order in time discretizations (unless you take very small timestep sizes for dG(0)). In the picture below you can see the dG(r) method with to time steps applied to the ODE $$\partial_t u = u$$ with the initial condition $$u(0) = 1$$. You clearly see that dG(1) and dG(2) are much closer to the analytical then dG(0) for only two temporal elements.
The closest answer that I know to your question is that different choices of basis and test functions are going to have different stability properties. In some cases, you can show that the Galerkin-in-time discretization is equivalent to some conventional differencing or Runge-Kutta scheme, which makes the analysis particularly simple. For example, CG(1) basis functions with DG(0) test functions is equivalent to the midpoint method, so it converges as $$\mathscr{O}(\delta t^2)$$ and has a favorable stability region for doing wave-type problems. Meanwhile using DG(0) basis and test functions is equivalent to the usual 1st-order backward scheme, which has a favorable stability region for parabolic problems.