Methods for designing difference schernes 221- Variational difference methods (the Ritz method and the Bubnov-Ga-
lerkin method). The Ritz and the Bubnov-Galerkin variational methods
have had considerable impact on con1plex numerical 111odeling problems and
designs of difference schemes.
Let A be a self-adjoint positive definite linear operator in Hilbert space
H equipped with an inner product (,) and let f be a given element of the
space ff. The problem of minimizing the functional
( 18) J[u] =(Au, 11.) - 2 (u, f)
is equivalent to the proble111 of solving the equation( 19) Au= f.
The element u 0 E H satisfying the equation Au 0 = f and realizing111in J[ u] = J[ u 0 ]. is umque..
The main idea behind the Ritz inethod is to take into consideration
a sequence of finite-dimensional spaces Vn with basis functions tp~n), i =
1, 2, ... , n, and look for an element un E Vn, minimizing the functional J[u]
in the space Vn.
Still using the framework of this method, we may attempt an approx-
imate solution 1in in the form
(20)n
Un = L Yj l.fj
j=l
with unknown coefficients y 1 , y 2 , ... , Yn. By inserting this express10n m
the formula for J[u] we find that
n n
(21)
i, j =l
where(22) j3i = (f, tp;).
Since A= A* is a self-adjoint operator, we have cxij = CXJ·i· The functional
J[ unJ is a function of n coefficients y 1 , y 2 , .•• , Yn. By equating the deriva-
tives oI[unJ/oy; to zero and using the symmetry of coefficients cxij = cxj;,
we obtain n equations for dete1·mination of Yi:
(23)
n
L (Xij Yj - /3; = 0,
j=li = 1,2, ... ,n.