1549301742-The_Theory_of_Difference_Schemes__Samarskii

Methods for designing difference schemes 225

we reduce the syste1n of equations (35) with the inembers &-;- and b.; to

(38)

&-(y·-Y·) &+(y· -y·)

+ i i h i-1 + i i+I h i _ d .i y,. = _. tp, '

i=l,2, ... ,JV-1.

Thus, we arrive at the difference scheme

0 < x = ih < 1'

(39)

•1.j ' 0 = (^0) ' YN = 0,
whose coefficients can be recovered from ( 30), ( 31) and ( ;3 7).
For r(x) 0 this scheme is identical with sche111e (31)-(32) obtained
by means of the Ritz n1ethod. In the case of constant coefficients k(:r ), r(:r)
and q(x)
h2
a; = k - B q, d; = d = q,
r
b~ ' = &+ z = -, 2 b~Yx-z + &+yx ' = ryo x.
When the coordinate functions tp; ( x) = 17 ( ( x - X;) / h) are chosen by
an approved rule as suggested before, the Ritz and the Bubnov-Galerkin
methods coincide with the finite element method.

4. The method of approximating a quadratic functional. The boundary-
   value problem

Lu= (ku')' - q(x)u = -.f(x), O<x<l, u(O)=O, u(l)=O,

is equivalent to the problem of searching a minimizing element for the

functional (see Section 3)

1

( 40) J[u] = j [k(u')^2 + qu^2 ]

()

1

dx - 2 j .f u dx.

0

Recall that the equation Lu= -.f(x) is Euler's equation related to such a
functional J[ u].