# Non-stationary Stokes on closed surfaces

I have made a solver for non-stationary (generalized) Stokes on surfaces. I also have a solver for stationary Stokes on surfaces. These are both based on Lehrenfeld/Lederer/Schöberl 2020.

My problem concerns solving the non-stationary on a sphere. My stationary solver works fine. From a solution (u,p) for the stationary Stokes with right-hand side “force” and div(u) = g. I have constructed a non-stationary solution (U,P) to non-stationary stokes with right-hand side (Force, G) by setting

U = w(t) * u
P = w(t) * p
Force = w(t) * force + w_t(t) * u
G = w(t)*g.

However, my solution explodes, even if I set w(t) = 1, w_t(t) = 0. Note that it works fine, and converges, for open surfaces (with Dirichlet boundary conditions).

For clarity, the time-stepping is performed with Implicit Euler, and the mixed space is a Piola-transformed BDM_k, P^{dc}_{k-1}. I also have a Lagrange-multiplier for uniqueness of p. (Also I do Hybridized Hdiv).

Qualitatively, my solution is continuously perturbed for a few time-steps to what looks like another smooth solution. After that, it just increases in magnitude. The factor with which it increases in each step seems to be independent of time-step size. It almost seems like there is another constant (in time) term on the right-hand side, independent of time-step size.

Does anyone know why this problem arises? Is there something about non-stationary stokes on closed surfaces which is ill-posed?

I would be very grateful for any insight Best regards,

Alvar

Update:
It turns out that since when there are no BC I do not need to homogenize, I forgot to reset the new solution. This has the “explody” nature a bit. For this discussion I have w(t) = 1, w_t(t) = 0.
For dt<= 2.e-9 the system seems to be stable, i.e. it only changes mariginally in time. However, for dt>= 3.e-9 it seems to increase steadily, and at e.g. 1.e-5 we are at full “explosion” territory.

It also explodes less when I remove the Lagrange-multiplier for p.