div_div(3rheolef) [debian man page]

div_div(3rheolef)						    rheolef-6.1 						 div_div(3rheolef)

NAME

       div_div -- -grad div operator

SYNOPSIS

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

DESCRIPTION

       Assembly the form associated to the grad(div(.))  operator on the finite element space V:

			/
			|
	       a(u,v) = | div(u) div(v) dx
			|
			/ Omega

       This form is usefull when considering elasticity problem.

EXAMPLE

       Here is an example of the vector-valued form:

	       geo omega ("square");
	       space V (omega, "P2", "vector");
	       form  a (V, V, "div_div");

rheolef-6.1							    rheolef-6.1 						 div_div(3rheolef)

s_curl(3rheolef)						    rheolef-6.1 						  s_curl(3rheolef)

NAME

       s_curl -- curl-like operator for the Stokes stream function computation

SYNOPSIS

	     form(const space M, const space& V, "s_curl");

DESCRIPTION

       Assembly the form associated to the s_curl operator on a finite element space V:

		       /
		       |
	     b(xi,u) = |  u.s_curl(xi) dx
		       |
		       / Omega

       The  M  and  V  space  may be a either P1 or P2 finite element space.  The M space is scalar-valued while the V is vector-valued.  See also
       form(2) and space(2).

       For cartesian coordinate systems, this form coincide with the usual "curl" one (see curl(3)).  In the axisymmetric case:

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

       The b form is denoted as "s_curl", for Stokes stream function computation (see s_grad_grad(3)) as it is closely related to the "curl" oper-
       ator (see curl(3)), 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  )

       Notice  also that the differentiation is performed on the xi variable here: b(xi,u)=(s_curl(xi),u) while the "curl" form brings the differ-
       entiation on the u vector-valued variable: (curl(u),xi), i.e. a transpose formulation.

ORIENTATION AND SIGN FIX

       The (r,theta,z) coordinate system has positive orientation, thus (z,r,theta) and (z,r) are positive also.  But (r,z,theta)  and	(r,z)  are
       negative : the sign of s_curl is then inverted to obtain the same result as if (z,r) was used.

EXAMPLE

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

	       geo g("square");
	       space M(g, "P1");
	       space V(g, "P1", "vector");
	       form a(M, V, "s_curl");

SEE ALSO

       form(2), space(2), curl(3), s_grad_grad(3), curl(3)

rheolef-6.1							    rheolef-6.1 						  s_curl(3rheolef)