# Can Mathematica's FEM solve coupled Dirichlet Boundary Conditions?

I am solving a coupled system of PDEs using Mathematica's FEM capabilities.

Specifically, the Navier-Stokes equations with a no-flux stress-free boundary.

To do this, I need to specify a Dirichlet boundary condition that looks like

n.u = nx ux + ny uy = 0 on the boundary.

But in Mathematica this gives the warning

NDSolveValue::fembdcc: Cross-coupling of dependent variables in DirichletCondition[ ux+uy==0,x+y==0] is not supported in this version.

This is similar to a previous question. That answer gave a workaround using an added low viscosity shell matching onto a no-slip boundary. I would like to avoid this kludge.

Is there any to solve coupled Dirichlet boundary conditions in Mathematica 13.3.0?

Thanks

Eric

• Two comments: to me this looks more like Neumann value, but I could be wrong. 2) one could use low level FEM to add Lagrangian multipliers to model the cross coupling, if this truly a Dirichlet condition. Jun 29, 2023 at 20:05
• @user21: In Navier-Stokes one is solving for the velocity field $\vec{u}$, so Dirichlet BCs involve the values of $\vec{u}$ and Neumann BCs would involve the derivatives of $\vec{u}$. Jun 29, 2023 at 20:39
• @MichaelSeifert 'ux' looks a bit like a derivative but since there is no code it is hard to say. Jun 30, 2023 at 4:35
• @user21: I think it's supposed to be the $x$-component of $\vec{u}$. Requiring $\hat{n}\cdot \vec{u} = n_x u_x + n_y u_y = 0$ would then make sense as a "no-flux"boundary condition. Jun 30, 2023 at 12:06
• Michael is correct here. @user21 How would the Lagrangian multiplier work? Any example showing coupled Dirichlet conditions would be really helpful (e.g. 1D Poisson problems with coupled values). Jun 30, 2023 at 13:40