# Adaptive Piecewise Poly-Sinc Methods for Ordinary Differential Equations
## 4 Adaptive Piecewise Poly-Sinc Algorithm
### Error Analysis
Figure 15 shows the set \(\mathsf{R}\) for \(m=2N+1=5\) Since points and \(m=2N+1=7\) Since points. It is observed that increasing the number of Sinc points per partition leads to faster convergence and a fe

\[\max_{x\in[a,b]}|y(x)-y_{\mathrm{c}}^{(i)}(x)|\leq K_{i}\,E_{i}+\delta\bigg{(} \frac{1}{\pi}\ln(\overline{\overline{m}})+1.07618\bigg{)},\]5 (**a**) Fitting the upper bound (24) with the set \(\mathsf{R}\). (**b**) Visualization of the residual, absolute local approximation error, and the mean value. **Example 8**. : _We study the following BVP [28; 64; 65]_

\[\max_{x\in[a,b]}|y(x)-y_{\mathrm{c}}^{(i)}(x)|\leq K_{i}\,E_{i}+\delta\bigg{(} \frac{1}{\pi}\ln(\overline{\overline{m}})+1.07618\bigg{)},\]6

_with boundary conditions \(y(0)=y(1)=0\), where \(\alpha>0\) and_

\[\max_{x\in[a,b]}|y(x)-y_{\mathrm{c}}^{(i)}(x)|\leq K_{i}\,E_{i}+\delta\bigg{(} \frac{1}{\pi}\ln(\overline{\overline{m}})+1.07618\bigg{)},\]7

_For large values of \(\alpha\), the BVP (26) has an interior layer close to \(\bar{x}\)[28]. The exact solution is given by_

\[\max_{x\in[a,b]}|y(x)-y_{\mathrm{c}}^{(i)}(x)|\leq K_{i}\,E_{i}+\delta\bigg{(} \frac{1}{\pi}\ln(\overline{\overline{m}})+1.07618\bigg{)},\]8

_We use the values reported in [64], i.e., \(\alpha=100\) and \(\bar{x}=0.36388\). This value of \(\bar{x}\) was chosen so that \(\lim\limits_{\alpha\to\infty}y(\bar{x}^{+})\approx 2\)[64]._

_We set the number of Sinc points to be inserted in all partitions as \(m=2N+1=7\). The stopping criterion \(\epsilon_{\rm stop}=10^{-12}\) was used. The algorithm terminates after \(\kappa=15\) iterations and the number of points \(|\mathsf{S}|=21\),469._

_Figure 16a shows the approximate solution \(y_{\rm c}^{(15)}(x)\). A proper subset of the set of points \(\mathsf{S}\) is shown as red dots, which are projected onto the approximate solution \(y_{\rm c}^{(15)}(x)\). We plot the statistic \(\omega_{i}\) as a function of the iteration index \(i\), \(i=2,3,\ldots,15\), in Figure 16b. One finding is that the oscillations are decaying and the statistic \(\omega_{i}\) is converging to an asymptotic value. The mean value \(\overline{\omega_{i}}\) is \(0.46\)._

_We perform the least-squares fitting of the logarithm of the set \(\mathsf{R}\) to the logarithm of the upper bound (24), where the parameter \(\delta\) is multiplied by \(10^{-12}\) and \(10^{-12}\delta=O(10^{-19})\). Figure 17a shows the least-squares fitted model (24) to the set \(\mathsf{R}\). The residual, absolute local approximation

\[\max_{x\in[a,b]}|y(x)-y_{\mathrm{c}}^{(i)}(x)|\leq K_{i}\,E_{i}+\delta\bigg{(} \frac{1}{\pi}\ln(\overline{\overline{m}})+1.07618\bigg{)},\]9 (**a**) The approximating polynomial \(y_{\rm c}^{(15)}(x)\). A proper subset of the set of points \(\mathsf{S}\) is shown. (**b**) Plot of the statistic \(\omega_{i}\). \((\longrightarrow)\,\overline{\omega_{i}}\approx 0.46\). \[\max_{x\in[a,b]}|y(x)-y_{\mathrm{c}}^{(i)}(x)| =\|y(x)-y_{\mathrm{c}}^{(i)}(x)\|_{1}\]0 Plotting the set \(\mathsf{R}\) for \(m=2N+1=5\) Sinc points (\(\bullet\)) and \(m=2N+1=7\) Sinc points (\(\blacksquare\)). error, and the mean value for the last iteration are shown in Figure 17b. Fine partitions are formed near \(x=\overline{x}\) due to the presence of the interior layer, as shown in the plot of the \(L^{2}\) norm values of the residual over the partitions. The mean value \(\overline{R_{13}}\) is below the threshold \(\varepsilon_{\rm stop}=10^{-12}\)._

_We compare the \(L^{2}\) norm of the approximation error of our adaptive piecewise Poly-Sinc method with other methods in \[\max_{x\in[a,b]}|y(x)-y_{\mathrm{c}}^{(i)}(x)| =\|y(x)-y_{\mathrm{c}}^{(i)}(x)\|_{1}\]1 Method [28] requires a parameter for the construction of refinement intervals. The \(L^{2}\) norm value of the approximation error is smaller than those reported in [28; 65]._

**Example 9**. : _We consider the BVP [66; 67]_

\[\max_{x\in[a,b]}|y(x)-y_{\mathrm{c}}^{(i)}(x)| =\|y(x)-y_{\mathrm{c}}^{(i)}(x)\|_{1}\]2

_with boundary conditions \(y(-1)=-2,\;y(1)=0\) and \(\epsilon>0\) is a parameter.