Numerically solve differential equation that has no solution I am not really experienced in the field of numerical differential equations. I am only studying existence and uniqueness results about PDEs at the moment. Now I was wondering what would happen, if one tries to solve a PDE numerically, although the PDE 1) has no solution or 2) the solution is not unique?

The Navier-Stokes-Equation in 3 dimensions comes to mind. How do physicists approach this equation? Do they work with numerical solutions although there is no proof, that there is a solution at all?

I really appreciate your help!

Best,

Luke
What happens depends strongly on the software to compute the solution, but in general if you don't have a proof of existence and uniqueness as well as a proof of convergence to the unique solution, you are on very shaky ground.

Let's consider $y' = f$ with $y(0) = 0$. This can be solved by an ODE solver, under some assumptions on $f$ (say, continuity). Now let's say $f$ is a delta function at a random location in the domain. A numerical solver will never find it, and the answer will always be wrong.

The next situation is non-uniqueness. I wrote a few solvers for a PDE which had many solutions. What we attempted was to compute them all, and then select a subset of the solutions based on some property we thought was relevant. (To be fair, what actually happened is we got into fights about what was relevant, and eventually we just modeled the phenomenon with a different PDE that did have a unique solution. I can honestly say that non-uniqueness was one of the biggest headaches I've had solving PDEs.)

I've never attempted to solve an equation which known to not have a solution, but I cannot imagine it turning out well. The best you can hope for is an error message, but you probably won't get it. My suspicion is that your numerical method will compute the solution to an equation which is in some sense "close" to the original, and then you'll be in a truly devastating situation where you have a bunch of garbage which you think is a solution.

Hopefully someone else can offer insight into Navier-Stokes; I've never worked on it so I can't help you there.
I am engineer and I made quite a lot of numerical simulations on PDEs. My experience does not result form mathematical insight, but by practical experince.
Engineers are interested in dayly work problems. We can assume that some kind of solutions exists: No such thin like a Dirac impuls somewhere. You need experience, some kind of feeling, which problems tend to fail and which are not problematic. The usual necessary mesh play an important role.
If I am unsure whether the solution is correct I tend to change something slightly: The geometry, the mesh, the boundary conditions. It may be happen that I got a very different solution. That means I am on a wrong way.
There are other situations where I know that I solutions seems to exist, lets say turbulent hydrodynamics. But I know that cannot come close to that solution with my algorithms. Then it is a useful idea to calibrate the solution against a real measuring. Then you may change something (say optimzing a geometry) an need only to take into account the changes against the calibrated situation.