Dot logic with grad and nabla_grad

variational formulation
1

How is this evaluated in Fenics

dot( grad(u), grad(v))

If u and v are vectors.

From tensors algebra, it should be:

$$\partial_{x_k} u_i \partial_{x_j} v_k$$

But I think what Fenics is doing:

$$\partial_{x_k} u_i \partial_{x_k} v_j$$

With grad, I think it is contracting on the second index.

If someone has experimented it before can tell what exact operation is done and we make sure if it’s what is expected

2

See: How ufl.nabla_grad works? - #2 by dokken

and grad docs

3

Thank you for your reply. From documentation it is clear that nabla_grad is the transpose of grad. But my question was not about that. I was asking about the dot operator with two grads.

Mathematically the dot between two second order tensors should be equivalent to the matrix multiplication. But what I think Fenics is doing

$grad(u) \cdot grad(v) = A \cdot B.T$

4

Dot products are described in detail in: Form language — Unified Form Language (UFL) 2025.1.0 documentation