174
{Curlqicos(j1tX3)} iEN* ;jEN
{ CurlriCOS(j1tX3)} iE[1,n]; jEN
constitutes an orthogonal basis of V and L2(Ql
ii)
the set {grad pi; iEN*} constitutes an orthogonal basis of:
v,;V) = { rpE V, rp = gradp, pEH 2 (Qx) }
Proof: The demonstration of this theorem essentially lies on the results concerning the
special base used for the resolution of the two-dimensional problem and on the results of
tensorial product of the Hilbert's space.
We note:
Vx = ( !/J E U(Qx)2 / div!/J E U(Qx) , curl!/J E U(Qx); !/J.n = ° on Yx}
We have U(Q)2 = U(Qx x ]O.lfY = U(O,1) @ U(Qx)2. Therefore, according to the
properties of the tensorial products of Hilbert's space, if (rpd is a base of U(O,1) and {lj/j} a
base of U(Qx)2, the set {rpilj/j} constitutes a base of U(Q)2.
Moreover, if {rpd is a base of Hi (O,l), and {lj/j} a base of Vx, then the set {rpilj/j}
constitutes a base of V 0' = HI (0,1;U(Qx)2)nU(0,1;Vx).
We therefore easily deduce from the theorem established for the resolution of the
Shallow water problem (Orenga [1992]) that the set:
{gradpicos(j1tX3);iEN*;jEN }u{ Curlqicos(j1tX3);iEN*,jEN }u{ Curiricos(j1tX3);iE f J,n]
,jE N} constitutes a base of V 0"
We also easily deduce from the previous result a base of Vu' For this we just have to
suppress the V 0' functions that do not verify the equation (III.35).
•
III.4.2 Properties of the special basis
We present hereafter, the main properties of the special base for the numerical
resolution of the sigma equations presented in the first part by the Galerkin's method. To
simplify the notations, we assume here a simply connected domain.