## SYNOPSIS

form(const space V, const space& V, "s_grad_grad");

## DESCRIPTION

Assembly the form associated to the -div(grad) variant operator on a finite element space V. The V space may be a either P1 or P2 finite element space. See also form(2) and space(2). On cartesian coordinate systems, the form coincide with the "grad_grad" one (see grad_grad(3)):

```               /
|
|
/ Omega

```
The stream function on tri-dimensionnal cartesian coordinate systems is such that

```       u = curl psi
div psi = 0

```
where u is the velocity field. Taking the curl of the first relation, using the identity:

```        curl(curl(psi)) = -div(grad(psi)) + grad(div(psi))

```
and using the div(psi)=0 relation leads to:

```        -div(grad(psi)) = curl(u)

```
This relation leads to a variational formulation involving the the "grad_grad" and the "curl" forms (see grad_grad(3), curl(3)).

In the axisymmetric case, the stream function psi is scalar ans is defined from the velocity field u=(ur,uz) by (see Batchelor, 6th ed., 1967, p 543):

```                 d psi                       d psi
uz = (1/r) -----    and   ur = - (1/r) -----
d r                         d r

```
See also http://en.wikipedia.org/wiki/Stokes_stream_function . Multiplying by rot(xi)=(d xi/dr, -d xi/dz), and integrating with r dr dz, we get a well-posed variationnal problem:

```        a(psi,xi) = b(xi,u)

```
with

```                  /
| (d psi d xi   d psi d xi)
a(psi,xi) = | (----- ---- + ----- ----) dr dz
| ( d r  d r     d z  d z )
/ Omega

```
and

```                /
| (d xi      d xi   )
b(xi,u) = | (---- ur - ---- uz) r dr dz
| (d z       d r    )
/ Omega

```
Notice that a is symmetric definite positive, but without the 'r' weight as is is usual for axisymmetric standard forms. The b form is named "s_curl", for the Stokes curl variant of the "curl" operator (see s_curl(3)) as it is closely related to the "curl" operator, but differs by the r and 1/r factors, as:

```                   (       d (r xi)     d xi )
curl(xi) = ( (1/r) -------- ; - -----)
(         d r        d z  )

```
while

```                   ( d xi       d xi )
s_curl(xi) = ( ----  ;  - ---- )
( d r        d z  )

```

## EXAMPLE

The following piece of code build the form associated to the P1 approximation:

```        geo g("square");
space V(g, "P1");