Equivalence between two systems of vector equations

Question

I need to solve the system

$$\nabla^2 \mathbf{u} = \nabla p \\ \nabla \cdot \mathbf{u} = 0$$

in a subdomain of $\mathbb{R}^3$ with mixed boundary conditions, where $\mathbf{u}$ is vector field and p is a scalar field. Taking the divergence of the first equation, using the definition of vector Laplacian $\nabla^2 \mathbf{u} = \nabla(\nabla \cdot \mathbf{u}) - \nabla \times (\nabla \times \mathbf{u})$, the second equation and the identity $\nabla \cdot (\nabla \times \mathbf{v}) = 0$, we have $$ \nabla \cdot (\nabla^2 \mathbf{u}) = 0 = \nabla \cdot \nabla p = \nabla^2 p$$

I would like to know if the system $$ \nabla^2 \mathbf{u} = \nabla p \\ \nabla^2 p = 0 $$

which was derived using $\nabla \cdot \mathbf{u} = 0$, is equivalent to the first one.

It has the advantage of allowing to solve first for $p$ and then for $\mathbf{u}$, in a decoupled fashion. Unfortunately, according to my numerical experiments and save numerical problems, they are not equivalent, but I don't see why the solution for the second is not a solution for the first one.

Answer 1

I'm not quite sure what you mean by "equivalent" but here is a possible issue:

You have shown $\nabla \cdot u=0 $ $\Rightarrow$ $\nabla^2p=0$, which is correct according to your system. However, the converse does not necessarily hold:

$\nabla^2 p=0$ $\Rightarrow$ $\nabla \cdot (\nabla^2 u) = 0$ which does not necessarily entail that $\nabla \cdot u=0$. In particular, any $u$ satisfying Laplace's equation satisfies this. So the equations $\nabla \cdot u=0$ and $\nabla^2 p=0$ are not "equivalent."

Answer 2

We're kind of going back and forth a lot, so I'll move it to an answer.

Typically when we want to solve unforced incompressible Stokes flow (I assume this is your task at hand?), to enforce incompressibility, we must solve the Poisson equation for pressure $p$ (your last equation), and use this result to plug in to the momentum balance (the first equation).

Of course, even being able to solve any of these depend on the well-posedness of the corresponding equation, and for the Poisson equation that means a well-defined boundary data. "Not knowing" your pressure along the walls means you need to make an ansatz about what happens here: I might suggest a linear interpolation between the inlet pressure and outlet pressure (choosing an appropriate ansatz is itself a complex issue). Whatever you do, you need boundary data for $p$ along the entire boundary in order to make the problem well-defined and to adequately solve the Poisson equation. Only then can you plug it in to the momentum balance to get your incompressible Stokes flow.