1549301742-The_Theory_of_Difference_Schemes__Samarskii

96 Basic Concepts of the Theory of Difference Schemes

the input data. The two major aims, improved accuracy and better stabil-

ity, have to be balanced against the need to avoid expending unnecessary

computational effort. In the next few sections the advanced theory in such

matters will be developed and presented in full details.

3. On the concept of well-posedness for a difference problem. There is
   another matter which is one of some interest. In conformity with statements
   of problems of mathematical physics, it is fairly common to call a problem
   well-posed if the following conditions are satisfied:
   (1) the problem is uniquely solvable for any input data from some class;
   (2) a solution of the problem continuously depends on the input data.
   Being concerned with a solution Yh and input data 'Ph of a difference
   problem depending on the grid step h as the parameter, we introduce the
   concept of well-posedness for a difference problem in a similar manner.
   Varying h we are in possession of two sequences of solutions {yh} and
   input data {'Ph}, thus causing not only a single difference problem, but
   also a family of problems depending on the parameter h. The concept of
   well-posedness is introduced for a family of difference problems (schemes)
   aslhl--;0.
   We say that a difference problem (scheme) is well-posed if for all
   sufficiently small I h I < h 0
   (1) a solution Yh of the difference problem exists and is unique for all
   input data 'Ph from some family;
   (2) the solution Yh continuously depends on 'Ph and this dependence is
   uniform in h.
   More precisely, the second condition in1plies that there exists a con-
   stant M > 0 independent of h such as for sufficiently small I h I < h 0 the
   inequality
   (18)

holds, where :fh is the solution of the problem with the input data cph, and
II · ll(h) and II · ll( 2 h) are suitable norms on the set of grid functions given
on the grid w h.
The property of the continuous dependence of the solution for a dif-
ference problem on the input data is expressed by inequality (18) and is
treated as the stability of the scheme with respect to the input data or
simply stability (see Section 4.2)

4. Stability, approxnnation and convergence. We now examine a continu-
   ous problem of the type

(19) Lu=f(x) for xEG, l u = μ( x) for x E f ,