1549301742-The_Theory_of_Difference_Schemes__Samarskii

242 Difference Schemes for Elliptic Equations

If, for instance, h 1 - = h 1 + = h 1 , then A~ v = A1 v = vx, x,, etc. The

procedure of writing out the arguments is somewhat cumbersome. This is

especially true for the case p > 2. To make our exposition more transparent,

it will be sensible to introduce the notations

x(+l,) = (x 1 + h 1 + ) x 2 ) )

x(±l^2 ) -- (x 1 ) x 2 ± h 2 ±) )

The disposition of the points x and x(±l.,) is shown in Fig. 9.

x

Figure 9.

The expression for A~ can be rewritten as

(14)

Ct=l,2.

In Chapter 2, Section 1 (see formula (27)) we have approved the expression

for vxx - v", whose use permits us to establish

(15)

Thus, on any irregular pattern the Laplace operator is approximated to
first order by the difference operator A* specified by formula (13).
The approximation of the type (12) applies equally well to a non-
equidistant grid and at the near-boundary nodes in the case of an arbitrary