1549301742-The_Theory_of_Difference_Schemes__Samarskii

244 Difference Schemes for· Elliptic Equations

By the same token,

UN~ II J \* UN= -^1

h

With these relations established, we arrive at the difference scheme

(17)

Yxx=-f(x),

A*y1=-J(x1),

xi = h 1 + ( i - 1) h,

A*yN = -f(xN), Yo = YN+ 1 = 0,

which will be convenient to be expressed with respect to the error z = y-u:

(18) A z = -1/J(x), 0 < x < 1,

where Az = z.-tx for x 1 <xi< xN, Az 1 = A*z 1 and AzN = A*zN, 1/Ji

O(h^2 ) for i = 2,3, ... ,N-l, 1/Ji = 0(1) for i = l,N.

In spite of the fact that this scheme generates no approximation at

the near-boundary nodes i = 1 and i = N, scheme (17) is of second-order

accuracy in the space C': II z lie = O(h^2 ). In order to obtain this estimate

at the points x = xJJ xN, we rewrite equation (18) as

where z 0 = hh 1 1/J 1 and zN+i = hh 2 1/JN· Thus, problem (18) is equivalent

to the following one:

Z;rx = -1/J(x),

, Zo = h h11/J1 '

On account of the a priori estimate

N i

llzllc<max(lzol,lzN+1l)+I: h L hl1/Jkl,

i=l k=l

arising in Section 2 of Chapter 1, it is easily verified that

This means that scheme (17) provides an approximation of order 2.