DG in time with a slab solver

Guosheng_Fu · April 20, 2021, 2:48pm

Hello,

I am exploring DG time solvers. In particular, I have a simple diffusion equation
u_t - u_xx=0.
I want to use CG to discretize in space and upwinding DG in time. I think the TensorProductFespace is designed for such problem, but I failed to produce a working code. (the tutorial examples in tensorproducts are all explicit DG…)

Mathematically, we shall be able to solve the linear systems in time slabs, I don’t know whether this would be possible to realize in the current TensorProductFEspace. Can you help me with this? Many thanks!

Best,
Guosheng

schruste · April 20, 2021, 6:34pm

As far as I know the TensorProductFESpace is not directly made for this.
In ngsxfem we have TensorProduct Space-Time spaces (also fitted), see e.g. here:

github.com

ngsxfem/ngsxfem/blob/release/demos/spacetime/spacetimeDG_fitted.py

"""
Heat equation (fitted) with Dirichlet b.c. solved with a DG-in-time
space-time finite element discretisation

Domain:
-------
The domain of this examples is [0,1]x[0,1]x[0,1] (2D+time interval).

PDE problem:
------------
  u_t - (u_xx + u_yy) = f in  Omega=[0,1]^2
                    u = 0 on dOmega
The r.h.s. term f is chosen according to a manufactured solution.

Discretisation:
---------------
* Discontinuous-in-time Galerkin space-time finite elements

Implementational aspects:
-------------------------------------------------------

This file has been truncated. show original

You can try it out without installation through binder here:

Best,
Christoph

Guosheng_Fu · April 21, 2021, 3:31am

Thanks for the examples.

Guosheng_Fu · April 21, 2021, 3:05pm

Hi Christoph,

Your spacetime framework is exactly what I am looking for. I have an additional question: I want to test ideas in 1D first as my model is a bit complicated. But your framework seems only support dimension d>=2. Is there a easy way around?

I am thinking of directing work on a tensor-product quad slab domain, but some of the finite elements seems missing: I need two nonstandard fe spaces: Pk-in-x direction and P-0 in y-direction (time), also Pk-in-x direction and P-k-1 in y-direction.
Are these spaces already available in ngsolve core?

Best,
Guosheng

schruste · April 21, 2021, 7:56pm

Hi Guosheng,

I just added 1D, at least for the scope of the examples that I’ve send you. You just need to update ngsxfem (master-branch). With that a script as attached works:
https://ngsolve.org/media/kunena/attachments/743/1d.py .

Best,
Christoph

Attachment: 1d.py