I’m trying to write a problem about the equation like below,

and I wrote:

problem displacementf(uf,duf)=int2d(th2)((dx(duf)*dx(uf)+dy(duf)*dy(uf))*uf)-int2d(th2)((dx(duf)*dx(uf)+dy(duf)*dy(uf))*convect([up1,up2],-dt,uf));

the procedure said the * is an wrong syntax.

I don’t know why.

is it a weak form of some pde?

also, what is duf and ((uf)_s)^’ ?

yes,it is a weak form. duf is the test function and equal to the partial(uf). (uf)'s=partial(uf)/partial(t)+vs*grad(uf). vs is the velocity.

If I am not wrong:

your governing PDEs is

uf_t + vs . ( grad(uf) ) + grad( uf ) * [ 0, wf ] - vf = 0

if Yes, then you need to linearized your problem first then you can solve it.

Also, if possible try to give the cartesian form of governing equation without weak formulation.

it is given here.

https://doc.freefem.org/tutorials/navierStokesNewton.html