Pressure integrated by parts in finite element method

Question

Most FEM texts or tutorials apply FEMs on diffusion equations where the 2nd spatial derivative is integrated by parts during weak formulation. For convection diffusion equations, there is a first spatial derivative term not integrated by parts. But in Navier-Stokes, $\nabla p$, a first spatial derivative, is integrated by parts, why?

Answer 1

The two weak formulations of the Stokes equation, $$\int_\Omega\bigl(\nabla\mathbf{v}\cdot\nabla\psi_i+\psi_i\nabla p \bigr) dV=0,$$ and $$\int_\Omega\bigl(\nabla\mathbf{v}\cdot\nabla\psi_i-p \nabla\psi_i\bigr) dV=0,$$ with $\psi_i$ a test function which vanishes on the boundary of $\Omega$, are both in use. (For the first formulation without partial integration of the pressure term, see Burkardt's lecture notes, section 4.)

The first formulation requires that both the velocity field and the pressure field must be $C^1(\Omega)$, in the second formulation the pressure can be $C^0(\Omega)$. It depends on application-specific reasons whether one requirement is favored over the other.