Let Ph : C([0, 1]) → F(ωh) be a so-called projection operator, which is defined
as (Phu)(xi) = u(xi) for any mesh-point xi ∈ ωh. We denote by eh ∈ F(ωh)
the function defined as eh = yh− Phu, which is called error function.
Then
eh(xi) = yh(xi) − u(xi) = yi− u(xi). (7.18)
Definition 7.1.11. The numerical method defined by the mesh-operator Lh is
called convergent in the maximum norm when lim
h→0kehk∞ = 0. (7.19)
If kehk∞ = O(hp) with some number p ≥ 1, then we say that the convergence
In the sequel we show that the numerical method (7.9) is convergent, and we define the order of the convergence, too.
Using the notation eh(xi) = ei, from (7.18) we have yi = ei + u(xi). By
substitution of this formula into the scheme (7.10), we get −ei+1− 2ei+ ei−1 h2 + cei = Ψ h i, i = 1, 2, . . . , N − 1, e0 = 0, eN = 0, (7.20) where Ψhi = fi+
u(xi+1) − 2u(xi) + u(xi−1)
h2 − cu(xi). (7.21)
Since fi = f (xi) = −u00(xi) + cu(xi), therefore
Ψhi = u(xi+1) − 2u(xi) + u(xi−1)
h2 − u 00 (xi) + (cu(xi) − cu(xi)) | {z } =0 . (7.22) Therefore, Ψhi = O(h2). (7.23) (In a more detailed form, by writing the leading constant, we have Ψhi = h2M
4/12 + O(h4), where the notation M4 = max[0,l]|u(4)| is used.) We denote
by Ψh ∈ F(ω
h) the mesh-function for which Ψh(x0) = Ψh(xN) = 0, and at
mesh-points of ωh it takes the values defined in (7.21). Then kΨhk∞= O(h2).
We notice than the error equation (7.20) can be written in the form
vheh = Ψh, (7.24)
where vh is the matrix of the form (7.12), eh denotes the vector from RN +1
which corresponds to the error function eh. Since vh is regular, we may write
eh = v−1h Ψh. Hence, using the inequalities (7.17) and (7.23), we get
kehk∞≤ kv−1h k∞kΨhk∞≤ K · O(h2) = O(h2). (7.25)
So limh→0kehk∞= 0. This proves the following statement.
Theorem 7.1.12. Assume that the solution of problem (7.1) u(x) is four times continuously differentiable. Then the numerical solution defined by (7.5)-(7.6) (or, alternatively, by the operator equation (7.9)) converges in second order in the maximum norm to the solution u(x).
Remark 7.1.1. In Theorem 7.1.12 the leading constant of the error function (i.e., the coefficient at h2) is M
Based on the relation (7.22), Ψh
i shows in what order the values of the exact
solution u(x) satisfy the numerical scheme at the mesh-points xi. Clearly, it is
a basic demand that with h decreasing to zero (that is, when – in some sense – the mesh ωh ”tends” to he interval [0, l]), then Ψhi should also tend to zero.
Hence, the following definition is quite natural.
Definition 7.1.13. We say that a numerical method with linear operator vh
is consistent when limh→0Ψhi = 0. When the method is consistent and Ψhi =
O(hq) (q ≥ 1), then the method is called consistent with order q.
According to our results, the method (7.9) is consistent with second order. As we have seen in the proof of Theorem 7.1.12, the consistency in itself is not enough to prove the convergence: we have required some other property, namely, the estimate (7.17), which will be defined in an abstract way, as follows. Definition 7.1.14. We say that a numerical method with linear operator vh is
stable, when the operators (matrices) vh are invertible, and there exists some
constant K > 0, independent of h, such that the estimate
kv−1h k∞≤ K (7.26)
is uniformly satisfied for any considered h.
Remark 7.1.2. For some numerical method the above stability property is valid only for sufficiently small values of h. This means that the there exists some number h0 > 0 such that the estimate (7.26) holds for any h < h0,
but not for any h > 0. For the first case (i.e., when (7.26) holds for any h < h0, only) the scheme is called conditionally stable. For the second case (i.e.,
when (7.26) holds for any h > 0) the scheme is called unconditionally stable. Since convergence examines the property of the scheme for small values of h, therefore the weaker property (the conditional stability) is enough (at least, theoretically) for our purposes. However, when h0 is extremely small, (which
may happen, e.g., in stiff problems), this may cause very serious problems. (E.g., when h0 is close to the computer zero.)
The following statement establishes the connection between the notions of consistency, stability and convergence.
Theorem 7.1.15. When a numerical method is consistent and stable, then it is convergent, and the order of convergence coincides with the order of consis- tency.
The above theorem plays a central role in numerical analysis, and it is applied in many aspects. Therefore sometimes it is referred to as the basic theorem of numerical analysis.
We enclose an interactive animation, called fdm eng.exe.The program can be downloaded from
http://www.cs.elte.hu/~faragois/nummod_jegyzet_prog/.
The image of this program on the screen can be seen in Figure 7.1.
Figure 7.1: The image on the screen of the interactive program for the numer- ical solution of the boundary-value problem by using finite difference method. Alternatively, this program suggests three initial-value problems as test problems. The chosen numerical method is the finite difference method. We can select the discretization step size by specifying the parameter n, which is the number of partitions of the interval. Pushing ”Calculate” we get the result. The result is given graphically, and the error (in the maximum norm) is also indicated. By increasing n the order of the convergence is also shown.