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2.3 Continuous and Discrete Solutions
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2.3 Continuous and Discrete Solutions
In the previous section, we saw that a finite difference scheme can produce
numerical solutions quite close to the exact solutions of two-point bound-
ary value problems. In this section, we shall go deeper into these matters
and show that almost all essential properties of the exact, or continuous,
solution are somehow present in the approximate solution. For this pur-
pose, we will need a bit more notation for the discrete solutions; in fact, we
find it useful to introduce a rather suggestive notation that can help us in
realizing the close relations. When this more convenient notation is intro-
duced, we will see that it is actually quite easy to derive properties such as
symmetry and positive definiteness in the discrete case simply by following
the steps of the proof for the continuous case. At the end of this section,
we will also prove that the finite difference solutions converge towards the
continuous solution as the mesh size h tends to zero.
2.3.1 Difference and Differential Equations
Let us start by recalling our standard two-point boundary value problem.
We let L denote the differential operator

(Lu)(x)= −u (x),

and let f ∈ C (0, 1) . Then, (2.1) can be written in the following form:

Find u ∈ C (0, 1) such that
2
0
(Lu)(x)= f(x) for all x ∈ (0, 1). (2.26)

Recall here that u ∈ C (0, 1) means that we want the solution to be
2
0
twice continuously differentiable, and to be zero at the boundaries. Thus,
we capture the boundary conditions in the definition of the class where we
seek solutions.
Now, let us introduce a similar formalism for the discrete case. First,
we let D h be a collection of discrete functions defined at the grid points
x j for j =0,... ,n + 1. Thus, if v ∈ D h , it means that v(x j ) is defined
for all j =0,... ,n + 1. Sometimes we will write v j as an abbreviation for
v(x j ). This should cause no confusion. Next, we let D h,0 be the subset of
D h containing discrete functions that are defined in each grid point, but
with the special property that they are zero at the boundary.
Note that a discrete function y ∈ D h has n + 2 degrees of freedom
y 0 ,y 1 ,... ,y n+1 . This means that we have to specify n + 2 real numbers in
order to define such a function. A discrete function z ∈ D h,0 has only n
degrees of freedom z 1 ,... ,z n , since the boundary values are known.
For a function w we define the operator L h by
w(x j+1 ) − 2w(x j )+ w(x j−1 )
(L h w)(x j )= − ,
h 2