2.5 Error Analysis
2.5.4 Convergence of the global solution
Proof. We know that LNej =τj. From Lemma 2.4.1, we have εkEk ≤C|τj| ≤C max
j=1,...,N
min
hj ε ,1
exp
−αxj−1 ε
+CN−1. (2.5.27) By using the inequality (2.5.26) in (2.5.27), we obtain the desired estimate.
Using Theorem 2.5.6, we can conclude that
|eu(x)−u(x)| ≤
CN−1+ Z xj
x |u0(s)|ds, ifk= 0, CN−1+C
Z xj
xj−1
|u0(s)|ds, ifk= 1.
Now we have to find the bound for the integral Ij =
Z xj
xj−1
|u0(s)|ds .
Using (2.2.7) and the equidistribution principle (2.3.2), we may conclude that Ij ≤ Cε−1
Z xj
xj−1
exp(−αs/mε)ds
≤ Z xj
xj−1
p1 + (u0(s))2ds= 1 N
Z 1
0
M(u(s), s)ds
≤ C`
N ≤CN−1. Hence, the desired result follows.
2.5.5 Error in the global normalized flux
The bound given in Theorem 2.5.11 provides the estimate for the pointwise error in the normalized flux defined by εu0(x). The following theorem proves the uniform convergence estimate for the global normalized flux.
Theorem 2.5.13. Let u(x) be the solution of (2.1.1) and u(x)e be the global solution defined as in (2.5.29). Then, the error of the normalized flux satisfies
ε|u0(xj)−ue0(xj)| ≤CN−1, j= 0, . . . , N. (2.5.30) Proof. For constant interpolant, we can get the required bound using Theorem 2.5.11. Now for the case of linear interpolant defined in (2.5.29), we have
ue0(xj) = UjN −UjN−1 hj . Using the Taylor series expansion, we obtain that
ε|u0(xj)−ue0(xj)| ≤ Cε hj
|u(xj−1)−Uj−1N |+|u(xj)−UjN|
+Cεhj|u00(ξ)|, (2.5.31) whereξ ∈(xj−1, xj). Now we distinguish the following two cases.
Case (i): hj = O(ε) (i.e., for the mesh points inside the layer region). Using the bound for the error given in Theorem 2.5.6, the inequality (2.5.31) can be bounded as
ε|u0(xj)−ue0(xj)| ≤CN−1+Cε2|u00(ξ)|.
Now using the bound ofu(x) given in Lemma 2.2.3, we can obtain the required estimate, i.e., ε|u0(xj)−ue0(xj)| ≤CN−1.
Case(ii): On the other hand, whenhj ≤CN−1(i.e.,for the mesh points outside the layer region), using Theorem 2.5.6, we obtain
ε|u0(xj)−eu0(xj)| ≤CN−1+CεN−1|u00(ξ)|.
Finally, using the assumptionε≤CN−1 and the bound given in Lemma 2.2.3, we can obtain the desired result.
2.6 Numerical Results
In this section to validate the theoretical results, the proposed numerical scheme is applied to several test problems with constant and variable coefficients having left and right boundary layers.
For comparison purposes, the numerical solution obtained by the upwind differences scheme on the piecewise-uniform Shishkin mesh are used.
Example 2.6.1. Consider the test problem
−εu00(x)−u0(x) +u(x) = 0, x∈(0,1), u(0) = 0, u(1) = 1.
The exact solution is given by
u(x) = exp(m1x)−exp(m2x)
exp(m1)−exp(m2) , where m1,2= −1±√ 1 + 4ε
2ε .
This BVP has a boundary layer in the left end at x= 0.
Example 2.6.2. Consider the variable coefficient problem
−εu00(x)− 1 +x(1−x)
u0(x) =f(x), x∈(0,1), u(0) = 0, u(1) = 0,
where f(x) is chosen in such that its solution u(x) is of the form u(x) = 1−exp(−x/ε)
1−exp(−1/ε) −cosπ
2(1−x) .
The above problem has a boundary layer at the left side of the domain near x= 0.
Example 2.6.3. Consider the variable coefficient problem
−εu00(x) + 1 +x(1−x)
u0(x) =f(x), x∈(0,1), u(0) = 0, u(1) = 0.
Now f(x) is chosen in such a way that its solution u(x) is of the form u(x) = 1−exp(−(1−x)/ε)
1−exp(−1/ε) −cosπx 2
.
The above problem has a boundary layer at the right side of the domain near x= 1.
For any value ofN andε, the exact maximum pointwise errorsENε and the corresponding rates of convergence are calculated by
ENε = max
0≤j≤N|u(xj)−UjN| and rNε = log2 EεN
Eε2N
,
whereuis the exact solution andUjN is the numerical solution obtained by usingN mesh intervals in the domain ΩN.
Now we would like to see the errors associated with the global solution and with the weighted derivatives. To do that, we calculate the maximum errors at the midpointsx∗j = (xj+xj+1)/2 of the corresponding adaptive mesh. The errors associated with the global solution and the corresponding rates of convergence are obtained by
EeεN = max
x∗j∈ΩN
ε|u(x∗j)−u(xe ∗j)| and erNε = log2 EeεN Eeε2N
! ,
whereu(x) is the exact solution andu(x) is the global solution as defined in (2.5.29). Similarly, wee can define the pointwise errors associated with the normalized flux as
DNε = max
1≤j≤Nε|u0(xj)−D−UjN| and pNε = log2 DεN
D2Nε
, and the global error for the normalized flux as
DeεN = max
x∗j∈ΩN
ε|u0(x∗j)−ue0(x∗j)| and peNε = log2 DeNε Deε2N
! .
Figure 2.3(a) represents the movement of mesh after each iteration and Figure 2.3(b) shows the final computed mesh corresponding to the solution of Example 2.6.1. It is clear from these two figures that the mesh starts to move towards the boundary layer and clusters as many points required for the layer region. Also we plot similar graphs in the case when the boundary layer is located at the right i.e., for Example 2.6.3. In Figure 2.6(a) the movement of the mesh towards right is presented and the final mesh is shown in Figure 2.6(b). These figures help us to conclude
that we can construct suitable non-uniform mesh through the same monitor function as defined in (2.3.3) even in the case of right boundary layer.
Figures 2.4(a) and 2.4(b) represent the global solution along with the exact solution and the corresponding error obtained on the adaptive grid for Example 2.6.2, respectively. Similarly Figures 2.5(a) and 2.5(b) show the normalized flux and the corresponding error.
In Table 2.1 and Table 2.2, the maximum pointwise error and the corresponding order of convergence for the solution and it derivatives respectively, are presented for Example 2.6.1. Similar results are shown in Table 2.3 and Table 2.4 for Example 2.6.2 which clearly show that the proposed method is ε-uniform convergent of order one. But for small values of ε, the rate of convergence becomes very slow due to the effect of high condition number of the coefficient matrix. As the condition number is directly proportional to square ofN and inversely proportional toε, hence for small values ofεthe condition number is very high resulting the coefficient matrix nearly singular.
One can refer [59] for more details on this argument. The same argument is also valid for the results in the subsequent chapters. At the same time if we useWjN (see [7]) instead ofUjN as given in (2.3.7), we can get better results for non-homogeneous problems which is shown in Table 2.9.
The computational results using the adaptive mesh are also compared with the numerical results using the Shishkin mesh which are shown in Table 2.5 and in Table 2.6 forε= 1e−3 andε= 1e−6 for Example 2.6.2. From these results, one can observe that the adaptive mesh produces errors almost of the same order as produced by using the Shishkin mesh. The advantage of this approach is that without any prior knowledge of the location of the boundary layer, we are able to generate an appropriate nonuniform mesh suitable for the layer type problems.