s_grad_grad(3rheolef) [debian man page]

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)

mass(3rheolef)							    rheolef-6.1 						    mass(3rheolef)

NAME

       mass -- L2 scalar product

SYNOPSIS

	       form(const space& V, const space& V, "mass");
	       form(const space& M, const space& V, "mass");
	       form (const space& V, const space& V, "mass", const domain& gamma);
	       form_diag(const space& V, "mass");

DESCRIPTION

       Assembly the matrix associated to the L2 scalar product of the finite element space V.

			/
			|
	       m(u,v) = | u v dx
			|
			/ Omega

       The V space may be either a P0, P1, P2, bubble, P1d and P1d finite element spaces for building a form see form(2).

       The  use of quadrature formulae is sometime usefull for building diagonal matrix.  These approximate matrix are eay to invert.  This proce-
       dure is available for P0 and P1 approximations.

       Notes that when dealing with discontinuous finite element space, i.e.  P0 and P1d, the corresponding mass matrix is block diagonal, and the
       inv_mass form may be usefull.

       When two different space M and V are supplied, assembly the matrix associated to the projection operator from one finite element space M to
       space V.

			/
			|
	       m(q,v) = | q v dx
			|
			/ Omega

       for all q in M and v in V.

       This form is usefull for instance to convert discontinuous gradient components to a continuous approximation.  The transpose  operator  may
       also be usefull to performs the opposite operation.

       The following $V$ and $M$ space approximation combinations are supported for the mass form: P0-P1, P0-P1d, P1d-P2, P1-P1d and P1-P2.

EXAMPLE

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

	       geo g("square");
	       space V(g, "P1");
	       form m(V, V, "mass");

       The use of lumped mass form write also:

	       form_diag md(V, "mass");

       The following piece of code build the projection form:

	       geo g("square");
	       space V(g, "P1");
	       space M(g, "P0");
	       form m(M, V, "mass");

SCALAR PRODUCT ON THE BOUNDARY

       Assembly  the  matrix  associated to the L2 scalar product related to a boundary domain of a mesh and a specified polynomial approximation.
       These forms are usefull when defining non-homogeneous Neumann or Robin boundary conditions.

       Let W be a space of functions defined on Gamma, a subset of the boundary of the whole domain Omega.

			/
			|
	       m(u,v) = | u v dx
			|
			/ Gamma

       for all u, v in W.  Let V a space of functions defined on Omega and gamma the trace operator from V into W.  For all u in W and v in V:

			 /
			 |
	       mb(u,v) = | u gamma(v) dx
			 |
			 / Gamma

       For all u and v in V:

			 /
			 |
	       ab(u,v) = | gamma(u) gamma(v) dx
			 |
			 / Gamma

EXAMPLE

       The following piece of code build forms for the P1 approximation, assuming that the mesh contains a domain named boundary:

	       geo omega ("square");
	       domain gamma = omega.boundary();
	       space V	(omega, "P1");
	       space W	(omega, gamma, "P1");
	       form  m	(W, W, "mass");
	       form  mb (W, V, "mass");
	       form  ab (V, V, "mass", gamma);

SEE ALSO

       form(2)

rheolef-6.1							    rheolef-6.1 						    mass(3rheolef)