What operators are present in the multi-dimensional elliptical PDE?

I'm having trouble understanding how the multi-dimensional elliptical PDE, detailed in Coefficients for Systems of PDEs, works. In this case, u is a vector, so how is the gradient of u defined? Is the result a tensor? If so, what is the circled-cross operator? It seems to take two tensors (c and grad(u)) and turn them into a vector?

Thanks

Answer 1

[SBarth]

I found the definition of ∇u in this example:

This is an $N = 3$ system. The gradient of ${\bf u}$ is given by

$$ \nabla{\bf u} = \left\{ \begin{array}{c} \frac{\partial u}{\partial x} \\ \frac{\partial u}{\partial y} \\ [\medskipamount] \hline \frac{\partial v}{\partial x} \\ \frac{\partial v}{\partial y} \\ [\medskipamount] \hline \frac{\partial\phi}{\partial x} \\ \frac{\partial\phi}{\partial y} \end{array} \right\} $$

Answer 2

[Alan Weiss]

The definition of the c input is detailed here. In short, the gradient of u is defined as a vector for each component of u separately, and the c(circlecross)u notation is defined in the link I gave.

Alan Weiss

MATLAB mathematical toolbox documentation

[SBarth]

Thanks Alan. I've come across that before, but it doesn't seem to explicitly define the circlecross operator. I've previously tried to work backwards from what's given, but I think there's too much ambiguity in not knowing either the circlecross or the grad(u) result.

Are you saying that grad(u) (say, in two dimensions) is equal to grad(u_1) + grad(u_2)?

[Alan Weiss]

I don't understand exactly what you mean. The equations at the very top of that page define explicitly and completely what the \nabla\cdot(c(circlecross)u) operator means, and that's all you really need.

For the inner details, remember, \nabla\cdot A = \partial A(1) / \partial x + \partial A(2) / \partial y. You can backtrack from there to see what the suggestive notation means.

Alan Weiss

MATLAB mathematical toolbox documentation

[SBarth]

Sorry, let me clarify: the links we posted explicitly specify the result of the operation ∇⋅(c⊗∇u), from which one can easily determine the definition of c⊗∇u.

However, I don't know the definition of either ∇u (when u is a vector) or the ⊗ operator. I'd need to know one of these in order to work backwards from the given definition, but unfortunately, I don't.