1549301742-The_Theory_of_Difference_Schemes__Samarskii

704 Methods for Solving Grid Equations

where

a+ Q =Cl Ct' (xC+lac)) l

l

Yx" = h+ (y(+lac) -y)'

°'

= _l (y'(+la) _ y).

Y.i" fi°'

Here the coefficients o.a and \0( x) are .so chosen as to provide on a uniform

grid a local approximation of order 2. By analogy with Chapter 4, Section

3 it is plain to justify the uniform convergence of scheme ( 81) with the rate

O(lhl^2 ).

We proceed to more a detailed exploration of MATM for solving the

systen1 (81) of difference equations just established. For this, the sum

A y = Ai y + A2 y involves the members

(82)

(83)

governing what can happen: Yx" = -

1

-y if xC-ta) E 'Yh and Yx = -

1

-y if

hcY " ha

xl+t,,)Eih·

Other ideas are connected with operators Ai and A 2 such that A" y =

0 0

-A", y, ct = 1, 2, for any y E SI = H, where SI is the set of all grid functions

vanishing on the boundary 'Yh· By the same token, A= Ai+ A 2 =-A.

0

Under the inner product .structure in the space H =SI

(y, v) = L y(x) v(x) fi 1 fi 2

.i·Ew1i

the operators Ai and A2 are nmtually adjoint to each other: (Aiu, v)
(y, A2v). Because of this fact, the operator A= Ai+ A 2 is self-adjoint.