# Space-time finite element discretization for time-dependent PDEs

In FEM literature, semi-variational methods are typically used in the solution of time-dependent PDEs. I have not seen a fully-variational approach i.e. where space and time are discretised by FEM, perhaps allowing the use of unstructured space-time meshes. Although timestepping methods may be easier to implement, is there a particular reason why space-time meshing is not viable? I imagine one has to tailor meshes to respect the physical properties of a given problem, but I am not certain.

• Spacetime discretization is definitely a thing. The main disadvantage is that you have to work on a domain of one higher dimension, but some people have done it, and even developed some specialized preconditioners for the spacetime linear systems that arise. One major advantage is that one can paralellize over time through parallel linear algebra, whereas traditional timestepping requires one time to be solved before the next, and so forth. – Nick Alger Mar 26 '15 at 7:14
• Are you referring to the family of methods where you discretise time into slabs which are then triangulated? If not, is it possible for you to find an example of what you have described above? – stephn28 Mar 26 '15 at 7:30
• With respect to completely unstructured meshes in time, I've heard people mention the idea many times, but don't have any references offhand. – Nick Alger Mar 26 '15 at 21:14
• That is what I am pursuing at the moment, hence my search for relevant literature. Thank you for the help! – stephn28 Mar 26 '15 at 21:19
• For deriving error estimators (to drive adaptivity), I highly recommend the article "An optimal control approach to a posteriori error estimation in finite element methods" by Becker and Rannacher, numerik.iwr.uni-heidelberg.de/Paper/Preprint2001-03.pdf – Nick Alger Mar 26 '15 at 21:23

Full space-time discretization of time-dependent partial differential equations is indeed a thing. If you use a structured mesh in time (in the sense that the time discretization does not depend on space) and appropriate choice of trial and test functions, you can fit several standard time-stepping methods (Crank-Nicolson, implicit Euler or some Runge-Kutta schemes) into a Galerkin framework, which gives an elegant approach for analysis. This is described, for example, in Thomée's book Galerkin Finite Element Methods for Parabolic Problems (Springer, 2nd ed., 2006) or Chrysafinos' and Walkington's paper Error estimates for the discontinuous Galerkin methods for parabolic equations, (SIAM J. Numer. Anal. 44.1, 349–366, 2006).

Using a fully unstructured mesh is less common, but can make sense for hyperbolic problems where you have a transport of information along characteristics. If you use a discontinuous Galerkin formulation, each space-time element only couples with the neighboring element via face terms (you have no global continuity requirements), and you can use a sweeping process to compute the solution by going from element to element along characteristics -- a sort of "oblique" time-stepping. Of course, this is much more difficult to implement, even if it does not require storing the full space-time mesh (which can be prohibitive). On the other hand, you gain the advantage of unstructured meshes of allowing local (adaptive) refinement and hence locally adaptive time-stepping. One of the earliest references I know is Hughes and Hulbert: Space-time finite element methods for elastodynamics: formulations and error estimates, Computer Methods in Applied Mechanics and Engineering 66(3):339-363, 1988. There's also a PhD thesis by Shripat Thite on Spacetime Meshing for Discontinuous Galerkin Methods.

Another context where I have seen this idea is in PDE-constrained optimization for parabolic problems. There you can formulate the first-order necessary optimality conditions as a coupled system of forward-backward equations, which you can interpret as the mixed formulation of a 2nd-order in time, 4th-order in space elliptic equation with initial-final (and boundary) conditions. By doing an adaptive space-time discretization of this coupled system, you can have an efficient one-shot approach for computing the solution, see Gong, Hinze, Zhou: Space-time finite element approximation of parabolic optimal control problems, J Numer. Math. 20(2):111-145 (2012).

• Christian, are the RK schemes you mention implicit as well? – Jesse Chan Apr 14 '15 at 17:52
• Yes, at least the ones I know are. – Christian Clason Apr 14 '15 at 18:55

There are more recent papers on Space-Time Methods. There's one from Steinbach, Space-Time Finite Element and another from Langer et. al, Space-Time Isogeometric Analysis all addressing Parabolic Evolution Problems. In both articles, they describe vividly the variational formulations but in different settings. As the titles suggest, the former uses FEM and latter IgA. I think this gives good information particularly on what you seek.

In the last chapter of the second edition of the monograph Numerical Mathematics,Quatteroni et. al, there's a section on Space-Time which might also be helpful especially with connections to the $\theta-$schemes.

Tensor product Space-Time implementation is very different from non-tensor based ones. The latter is a bit tricky especially for the FEM.