Confused about use of multiplication symbol (*) in some UFL examples

Question

Hi,

I am new to FEniCS and have been looking at the incompressible Navier-Stokes demo and am confused about this UFL code representing a term in the weak formulation:

inner(grad(u0)*u0, v)

I am likely missing something, but I can't see how this is equivalent to the following term (given previously in the example):

$$\langle\nabla u \cdot u, v \rangle$$

The FEniCS book seems to indicate that the multiplication operator in UFL is only valid between two scalars, a scalar and a tensor, a matrix and a vector or two matrices. (I interpret grad(u0)*u0 as multiplication between a tensor and a vector.)

This usage seems to be prevalent throughout other examples as well, so I'd appreciate it if someone could clear this up for me!

Answer 1

With u0 vector-valued, grad(u0) is matrix-valued, and grad(u0)*u is the "a matrix and a vector" case you mentioned, i.e. it's equivalent to dot(grad(u0), u).