1549301742-The_Theory_of_Difference_Schemes__Samarskii

The sun11narized approxiination 1nethocl 639

In concluding this chapter it will be sensible to introduce the space

0

rt of all grid vector-functions given on the grid wh and vanishing on the

boundary ih of the grid under the inner product structure

n

(y,v)=~(y^8 ,v^8 ), (y^8 ,v^8 )= ~ Y^8 (x)v^8 (x)h 1 h 2 ... hp.

s=l xEwh

Having involved the operators

Ct p 0

A;;: y = - ~ A~ 3 y. At y = - ~ A!.u y, y E rt,

p=l ;3=et

we are going to show that these operators A- and A+ are mutually adjoint
each to other:

(A-y, v) = (y, A+ v)

0

for all y, v E rt ,

if the n1atrix k = ( k:;~') is syn11netric, that is, under the condition of syrr1-

metry k~r;l = k';;. Indeed, k~o: = k!μ and, because of this, we might
have

p 0:

(A- y,v) = 0.5 ~ ~ [(k~μYx~ 1 Yxa) + (k~μYx~,vxJ]

o:=I p=I

p )J

= 0.5 ~ ~ ~ ~ [(k-Ry"° ap "'fl ,V-· 1 • o ) + (k-Ry!' Cl:'p •-fj ,V~ •''Cl )]

/3= l o:=p

)J p

= 0.5 ~ ~ [(k~o:Yxa1v,,,,J + (k~o:Yxa1Yx~)J

o:=I p=o:

p p

= 0.5 ~ ~ [(k!μv,r;~ 1 Yx 0 ) + (k!μv.,, 1 ,,Y,vJ]

o:= I p=o: