1549301742-The_Theory_of_Difference_Schemes__Samarskii

Higher-accuracy schemes for Poisson's equation

holds. This implies that

Scheme (9) is of fourth-order accuracy when u E C(^6 ),

f E C(^4 ) and condition (11) is satisfied.

295

It is worth noting here that on the square grid (h 1 = h 2 = h) this

condition is automatically fulfilled. A proper choice of <p guarantees the

sixth order of accuracy of scheme (9) on any such grid. Convergence of

scheme (9) with the fourth order in the space C can be established without

concern of condition ( 11). An alternative way of covering this is to construct

an a priori estimate for 11 A z 112 and then apply the embedding theorem (see

Section 4).

Let Slh be the space of all grid functions defined at the inner nodes

x E wh of the grid wh = {(i1 hi, i2 h2)}, 0 < iex <Net, hex Nex =lex, CY= 1, 2,

0

and let Slh be the space of all grid functions defined on the grid w h and

vanishing on the boundary ih. The accepted view is that an inner product

in the space Slh such as

N1-l N2-l

(y, v) = L L y(i 1 h 1 , i 2 h 2 ) v(i 1 h 1 , i 2 h 2 ) h 1 h 2

ii =l i2=1

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

:CEwh

and the operators A 1 and A 2 specified by the relations

~ 0
where Aex y = Aex y for all y E Slh, will complement special investigations.
~ 0
Here Aex really acts from Slh into Slh and is identical with Aex itself in Slh·
Therefore, A 1 and A2 are linear operators defined on Hh = Slh (they can be
0 0
treated also as operators from Slh into Slh C Slh). The domain and range
of these operators coincide with Slh = Hh.
Here the complete posing of problem (10) is concerned with an oper-
ator equation

(12)