1549301742-The_Theory_of_Difference_Schemes__Samarskii

Homogeneous difference schemes for the heat conduction 479

0 A

space SL H of all grid functions given on the grid wh and vanishing at

the points x = 0 and x = 1 and then define several inner products by

(y, v )*

N-l

L Yi vi Ii; ,

i=l

N-l

(y,v)= L Yivihi,

i= l

N

(y,v] = L Y;Vihi.

i=l

In this direction we refer to the operator A : H r--+ H with the values

Ay = -Ay = -(ay,,,)i: for any y EH.

We know that the operator A so defined is self-adjoint and positive definite:

A= A*> 0,

since (Ay, v) = (ayx, v:r;] = (y, Av). By the same token,

IY I II r· ~ -_ O<z<N m.ax I Y, ·I < ~ 2 (Y:r:,Y:r]^112 < 2 yfC;^1 (Ay,y) i;2 ·

By obvious rearranging of the problem (50) solution as

,;., .. -- v +·u' 1,

where v is a solution of the same problem with another right-hand side
<fJ = 'T/i: and w is a solution of problem (50) with the right-hand side </J = </J*.
As can readily be observed, scheme ( 50) is stable under the constraints

where h 0 = min hi.

l<i<N

1 1

Clo= 2-rllAll' llAll < ~;, 0

Other ideas are connected with a priori estimate ( 41) for v and esti-
mate ( 42) for w. When providing such manipulations and establishing the
relation