s_grad_grad(3rheolef)						    rheolef-6.1 					     s_grad_grad(3rheolef)

NAME

       s_grad_grad -- grad_grad-like operator for the Stokes stream function computation

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)):

		      /
		      |
	     a(u,v) = |  grad(u).grad(v) dx
		      |
		      / 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");
	       form a(V, V, "s_grad_grad");

SEE ALSO

       form(2), space(2), grad_grad(3), grad_grad(3), curl(3), s_curl(3)

rheolef-6.1							    rheolef-6.1 					     s_grad_grad(3rheolef)