1549301742-The_Theory_of_Difference_Schemes__Samarskii

Difference approximation of elementary differential operators 69

The next step is to introduce an operator L in the space H 0 and an

operator L1i carrying a grid function vh into a grid function L1i vh on the

grid wh (L1i: H1i f-+ H1i).

A grid function

where vh = P1iv, (Lvh = P1i(Lv) and vis an arbitrary function (vector,

element) of the space H 0 , is called the error of approximation of the

operator L by the difference operator L1i.

If II 1/J1i ll1i ----+ 0 as h ----+ 0, we say that the difference operator L1i

approximates the differential operator L.

Likewise, the difference operator L1i approximates the differential

operator L with order m > 0 if

(28)

or II L,, V1i-(Lvh llh < M [ h Im, where Mis a positive constant independent

of I h 1.

Remark 1 We give below several examples of projectors P1i onto the set

of grid functions:

(1) for a continuous function v(x), vh = P1i v(x) = v(x), x E wh;

(2)

x+h 1

vh = P1i v =

2

\ j v(t) dt = ~ j v(x +sh) ds

x-h -1

if v(x) is a ,5um1nable function, etc.

Remark 2 The length [ h I= (h; +· · ·+ h~)^112 of a vector h = (h 1 , .•• , hp)

with components h 1 , ••. , hp may be taken as the quantity involved in the

above definitions. Observe that approximations with respect to he,, o: =

1, 2, ... , p, may be different in order. If so, instead of (28) we might have

p

II L1i vh - (Lv),, ll1i < M ~ h7:"',

<>=1

Choosing among the numbers m 1 , ••• , mp the minimal one and denoting it
by m, we get estimate (28).