Introduction - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 23

4.2 A Difference Equation for the Global Error. Let

Lu = f in ,

and suppose

A h U = F
is the associated linear system over h with F i = f(z i ) . In addition, suppose that
for the points on the boundary,

U(z j ) = u(z j ) for z j ∈ ∂ h .

In view of the linearity of the difference operator, it follows that the global error
η satisfies

(L h η) i = (L h u)(z i ) − (A h U) i
= (L h u)(z i ) − f(z i ) = (L h u)(z i ) − (Lu)(z i )
=− r i ,

( 4 . 1 )

= Lu − L h u is the local error on h . Thus, η can be interpreted as the
solution of the discrete boundary-value problem

where r :

L h η =− r in h ,

( 4 . 2 )

η = 0 n ∂ h .

4.3 Remark. If we eliminate those variables in (4.2) which belong to ∂ h ,we
get a system of the form

A h η =− r.

Here η i = η(z i ) for z i ∈ h . This shows that

convergence is assured provided r tends to 0 and the inverses A − 1

η is the vector with components

remain bounded

h

as h →

0. This last condition is called stability . Thus, consistency and stability
imply convergence.
In order to illustrate the error calculation by the perturbation method of (4.1),
we will turn our attention for a moment to a more formal argument. We investigate
the differences between the solutions of the two linear systems of equations

Ax = b,
(A + F)y = b,

where F is regarded as a small perturbation. Obviously, (A + F )(x − y) = Fx.
Thus, the error x − y = (A + F) − 1 Fx is small provided F is small and (A + F) − 1
is bounded. - In estimating the global error by the above perturbation calculation,
it is important to note that the given elliptic operator and the difference operator
operate on different spaces.
We will estimate the size of the solution of (4.2) by considering the difference
equation rather than via the norm of the inverse matrices A − 1

.

h