1549301742-The_Theory_of_Difference_Schemes__Samarskii

78 Basic Concepts of the Theory of Difference Schemes

corresponding to different values of the parameter his of great interest and

refers to a difference scheme.

The main goal of any approximate method is to solve an original

(continuous) problem with a prescribed accuracy E > 0 in a finite number

of operations. In order to clarify whether it is possible in principle to

approximate a solution u of problem (35)-(36) by a solution yh of problem

(37) with any prescribed accuracy E > 0 depending on the step h(E), we

follow established practice. This is concerned with further cornparison of

Yh with u(x) in the space of grid functions }fh. Let uh be a value of the

function u(x) on the grid wh, so that uh E Hh. The error zh = Yh - uh of

the difference scheme (37) needs more a detailed exploration for a complete

and rigorous treatment.

The condition for zh can be derived upon substituting Yh = zh + ·uh

into (37). Through such an analysis the problem of the same type arises as

problem (37):

(38)

where 1/Jh and vh are residuals equal to 1/Jh ='Ph -Lh uh and vh = xh -lh uh.

The right-hand sides 1/Jh and vh of problem (38) are called the error of

approximation of equation (35) by the difference equation (37) and the error

of approximation of condition (36) by the difference condition lh Yh = xh on

a solution of problem (35)-(36) or, briefly, 1/Jh is the error of approximation

for the equation Lh Yh = 'Ph on a solution u(x) to equation (35) and vh

is the error of approximation for the condition lh Yh = xh on a solution of

problern (35)-(36).

The accurate account of the error zh of a scheme as well as of the

errors of approximations 1/Jh and vh is carried out in suitable norms II· ll(h),

II · ll(h) and II · llc 3 h) on the space of grid functions.

We say that a solution of the difference problem (37) converges to a

solution of problem (35)-(36) (scheme (37) is said to be convergent) if

or
II zh ll(h) =II P(I h I) II, where P(I h I) --+ 0 as I h I--+ 0.
The difference scheme (37) is said to be convergent with the rate 0(1 h In)
or is of the nth accuracy order (of accuracy 0( I h In)) if for all sufficiently
small I h I < h 0 the inequality