1549301742-The_Theory_of_Difference_Schemes__Samarskii

280 Difference Schemes for Elliptic Equations

where

ACYy= -A-;,y= -2yxu/hCY

ACY Y = -Aty = 2 Yxa/hCY

for

for x CY = (^0) l
With this in mind, problem (8) is convenient to be taken in the operator
fonn A y = A y. The operator A so defined is self-adjoint and nonnegative:
A= A*,
where 6. =Amax= 4(h;^2 + h;^2 ).
Observe that the operators A 1 and A 2 are commuting and self-adjoint
both for the first and second boundary-value problems. The methodology of
the general theory (see Chapter 2, Section 1) guides the choice of a common
system of eigenfunctions coinciding with the system of eigenfunctions for
the operator A = A 1 + A 2 ; in so doing, each eigenvalue of the operator A is
equal to a sum of the appropriate eigenvalues of the operators A 1 and A2:

3. The Laplace operator in a domain consisting of rectangles. We now
   consider a domain consisting of a finite number of rectangles, whose sides
   are parallel to the coordinate axes as shown in Fig. 14.

/2 ~ - - - I- - - -1-- - -I - - - -..---....----.---...---...-----.

f-- - - - I-I - - -1-I - - -I I - - - +----t----t----t---+----1

~ - - -1-I - - -1-I - - -1-I - -

1--- - - I - - - - I - - - -I - - - -+----+----t----t----t----1 G

I I I

0 l l x l

Figure 14.