1549301742-The_Theory_of_Difference_Schemes__Samarskii

Stability and convergence of the Dirichlet difference proble1n 267

has been derived in Section 2 and may be useful in such a setting.

Having decomposed the right-hand side <pas

0

'P = 'P + <p*'

where~ = <p and <p* = 0 at the strictly inner nodes x E ~h (see Section 1)

0

and <p = 0 and <p* = <p at the near-boundary nodes x E wj,, we agree to

consider

y=v+w

with v and w being solutions of the problen1s

(7)

(8)

0

Av=-'P

Aw= -<p *

for x E wh,

for x E wh,

v[ -Yh = 0,

w[ -Yh = 0.

We are going to evaluate separately each of the functions v( x) and
w( x). In order to estimate v( x), it is necessary to construct a major ant
Y ( x). Assuming that the origin is inside the domain G, we try to determine
a majorant of the type

Y(x) = K (R^2 - r^2 ),

p
'\"""' x 2
a=l L., °''
where J( > 0 is a constant and R is the radius of a p-dimensional ball (a
circle for p = 2) with center at the origin containing entirely the domain
G. The constant K will be chosen a little later.
By virtue of the relations Aa x~ = 0 for CY -::/-/3 and

2 (x°' + hc,)

(^2) - 2 x (^2) + (xo: - hcx) 2
Ax °' °' = h2 °' =2 '
A x^2 =
°' °'
e - hex+ + hex-
°' - 2h°'
we find that
p
°'
AY = L AcxY = -2pf(
AY = -2pB J(
for
for x E w~,