1549301742-The_Theory_of_Difference_Schemes__Samarskii

254 Difference Schemes for Elliptic Equations

where 1jJ is the approxirnation error equal for cp(x) = f(x) to

1/;=Au+cp=Au-Lu at the regular nodes,

(28)

1/;* = A*u - Lu at the irregular nodes.

Let u E C(^4 l(G), where C(^4 ) is the class of functions u(x) with four

continuous in G derivatives with respect to x 1 , .•. , xp. As stated in Sec-

tion 3, we have at the regular nodes

(29) 1 ,1,l<M '// - 4 ihi2 12 ' I h^12 = h2 l + h2 2 + ... + h2 p.

Furthermore, in giving the approximation error at the irregular nodes as a

sum

p

(30) 1/J* = 2= 1/J~ ,

a= l

we apply the results obtained in Section 2 to the current situation:

(31) t/!* = 0(1)'

meaning that at the irregular nodes the scheme does not approximate the
equation i3.u + f(x) = 0.
Thus, in the p-dimensional case a difference scheme such as

p

Ay = L AaY = -f(x) at the regular nodes,

p

A*y=LA~y=-f(x) at the irregular nodes,

a=l

where Aa y = Yx" x" and A~ is specified by the formula

is associated with problem (1).