Introduction - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 22

§ 4. A Convergence Theory for Difference Methods

It is relatively easy to establish the convergence of finite difference methods, pro-
vided that the solution u of the differential equation is sufficiently smooth up to
the boundary, and its second derivatives are bounded. Although these assumptions
are quite restrictive, it is useful to carry out the analysis in this framework to
provide a first impression of the more general convergence theory. Under weaker
assumptions, the analysis is much more complicated; cf. Hackbusch [1986].

Consistency

In the following we shall write L h for the difference operator (which also specifies
the method). Then given u ∈ C() , L h u is a function defined at all points in h .
The symbol A h will denote the resulting matrix.

4.1 Definition. A finite difference method L h is called consistent with the elliptic
equation Lu = f provided

Lu − L h u = o( 1 ) on h as h → 0 ,

C 2 ( ¯ ) . A method has consistency order m provided that
for every u ∈ C m + 2 ( ¯ ) ,

for every function u ∈

Lu − L h u = O (h m ) on h as h → 0 .

The five-point formula (3.1) for the Laplace operator derived by Taylor ex-
pansions has order 1 for an arbitrary grid, and order 2 when the four neighbors of
the center point are at the same distance from it.

Local and Global Error

The definition of consistency relates to the local error Lu − L h u . However, the
convergence of a method depends on the global error

η(z i ) : = u(z i ) − U i

as z i runs over h . The two errors are connected by