Where u(x,z,t), \varepsilon(x,z,t), \sigma(x,z,t) and P(x,z,t) are the unkowns and the rest are constants. I want to solve this on a 2D cartesian grid.

I wonder what packages / methods I should look for in trying to solve this nonlinear problem. Are there any examples of similar things I can follow? Should I try PINNS or it would need much more than a single laptop?

Hi, this is a nonlinear Stokes PDE problem that is well-suited to be solved with Gridap.jl.

Perhaps you can take a look at the incompressible Navier-Stokes tutorial -also the p-Laplacian and elasticity/damage tutorials- to work out how to implement it. If you struggle to implement the nonlinear constitutive sigma-eps relation (I believe this could be the hardest part), you can try getting help in the gitter channel.

I would also take a look at Ferrite.jl. I am not familiar with the package, but maybe it can also solve this type of problem.

Hi, you could give Finite-Difference stencils a try on xPUs using ParallelStencil.jl, and potentially the accelerated pseudo-transient method (this paper and related 2D Stokes code) if you seek for fast steady state solutions.