1549301742-The_Theory_of_Difference_Schemes__Samarskii

Higher-accuracy sche1nes for Poisson's equation 297

it being understood that for any steps h 1 and h 2 the solution of problem

(12) can be estimated by

llzllc = lly-ullc < MllVill,

where

The estimate so constructed implies the uniform convergence of scheme (9)

with the rate O(lhl^4 ) for any ratio h 1 /h 2.

3. The multidi1nensional case. The method for constructing a scheme of
   fourth-order accuracy described in Section 1 applies equally well to the case
   of several variables, making it possible to compose the difference scheme of
   fourth-order approximation

(13) A'y = -\O(x), xEwh, y=μ.(x), J:E/h,

p h2 1-0-p

A'y = Ay + L l~ L Aex Aμ y,

cx=l poJcex

p

(14) Ay = L An Y,

ex= l

p h2

\0 = J + L 1 ; Aex J ,

ex=l

associated with problem ( 1) of Section 1 in the p-dimensional parallelepiped
Go= {O < Xex <lex, CY= 1,2, ... ,p} on the grid wh = {x; = (il hl , ... ,
ip hp), i ex = 0, 1, 2,... , N ex , hex N ex = lex}.
By introducing the space Hh = s:th and the operators Aex by analogy
with the preceding,section we draw the conclusion that the operator

p p

(15) A' = A - L XLY L Acx A13 ,

is self-adjoint and possesses the estimates

4 - p I

--A<A 3 - <A, -

h2

x = _!!._

ex 12 '

p > l.