The summarized approximation method 623The second con di ti on is ensured if
0
l!Aa μ B-^1 1/1 ~II = 0(1) as T ----> 0 , !hi ----> 0.The second method of special investigations with concern of additive
schemes was demonstrated in Section 8 in which convergence in the space
C of a locally one-din1ensional scheme associated with the heat conduction
equation was established by nleans of this method. Let us stress that in such
an analysis we assume, as usual, the existence, uniqueness and a sufficient
srnoothness of a solution of the original multidirnensional problem under
consideration.
Let, for example, ll be a solution of problern (5) and y = Yh be a
solution provided by this or that additive scheme, Yh E Hh, where Hh is
the set of grid functions. Following established practice, the difference z{ =
y{, - u{, where uh = Ph ll and Ph is a linear operator from the space H 0 into
the space Hh (u E H 0 , uh E Hh), needs investigation. To be more specific,
we are interested in the possible estimates of the quantity 11:0, - llf, ll(Jh)
in some suitable norm II · l!(h) on the space H h. The traditional ways of
covering this are to set up the relevant problem for the error zh, calculate the
0
residual Via = 1/1 a +Vi: and then adapt one of the well-developed methods
for det<-ermination of z h.- An approximation of the "1nultidiniensional" abstract. Cauchy proble1n
by a chain of the "one-diniensional" Cauchy proble111s. It is to be hoped
that the forthcoming reduction helps clarify what is done. The problem
staten1ent involves problem (5) with the homogeneous boundary conditions
on the boundary r under the agreement that the function u( x, t) as a
function of the argument x can be treated in a common setting as an element
of some vector normed space H 0. Then L refers to a linear operator in that
space and u = u(t) may be viewed as an abstract function of the argmnent
t with the values H 0 , it being understood that u( t) E Ho for all t E [O, t 0 ].
In this view, it is possible to write in problem (5) the usual derivative int
in place of the partial derivative, making our exposition more transparent.
As a final result we get the abstract Cauchy problem
(51)du
dt+Au=f(t), u( 0) = u 0 E Ho ,where A is a linear operator in the Banach space H 11. The don1ain D(A) C
H 0 of the operator A is everywhere dense in the space Ho and corn prises
all the functions satisfying the h01nogeneous boundary conditions on the
boundary rand its range 6.(A) belongs to the space H 0.