1549301742-The_Theory_of_Difference_Schemes__Samarskii

298 Difference Schemes for Elliptic Equations

These properties are an immediate implication of the chain of the relations

For p > 4 the difference operator (15) lacks the property of having fixed

sign (ellipticity). In each such case it is recommended to refer to another

operator A', which preserves the ellipticity property for any p:

or

p p

( 16) A' Y = L II ( E - xf3 Ap) Acx y.

It is evident that the approximation order remains unchanged and A' rs
identical with operator (15) up to the terms 0(1h1^4 ).
On the other hand, since E - xf3 Af3 > ~ E,

A I > LP Acx (2)P-l - -- (2)p-l - A

a= l^3 3

and, hence,

(~)p-

1

A<A'<A.

With the aid <;if the above operator inequalities we are able to produce
the necessary a priori estimates and justify the convergence with the rate
O(I h 14 ) for the scheme in hand. Observe that for p = 2 operator (16)
coincides with operator (15).
We have nothing worthwhile to add to such discussions, so will leave
it at this.