Theory and Modern Applications
Table 2 Numerical results with \(\varepsilon _{2}=2^{-9}\) for Example 1
\(\varepsilon _{1}\) |  | N = 32 | N = 64 | N = 128 | N = 256 | N = 512 | N = 1024 |
|---|---|---|---|---|---|---|---|
20 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.02e–02 | 2.74e–02 | 1.37e–02 | 7.30e–03 | 3.80e–03 | 1.90e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.87 | 1.00 | 0.91 | 0.94 | 1.00 | Â | |
K | 4 | 4 | 3 | 3 | 3 | 3 | |
2−2 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.26e–02 | 2.90e–02 | 1.45e–02 | 7.80e–03 | 4.10e–03 | 2.10e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.86 | 1.00 | 0.89 | 0.93 | 0.97 | Â | |
K | 4 | 4 | 3 | 3 | 3 | 3 | |
2−4 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.33e–02 | 2.94e–02 | 1.48e–02 | 8.00e–03 | 4.10e–03 | 2.10e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.86 | 0.99 | 0.89 | 0.96 | 0.97 | Â | |
K | 4 | 4 | 3 | 3 | 3 | 3 | |
2−6 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.11e–02 | 2.85e–02 | 1.44e–02 | 7.70e–03 | 3.90e–03 | 2.00e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.84 | 0.98 | 0.90 | 0.98 | 0.96 | Â | |
K | 4 | 4 | 3 | 3 | 3 | 3 | |
2−8 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 4.76e–02 | 2.62e–02 | 1.31e–02 | 6.80e–03 | 3.50e–03 | 1.80e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.86 | 1.00 | 0.95 | 0.96 | 0.96 | Â | |
K | 4 | 4 | 3 | 3 | 3 | 3 | |
2−10 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 4.70e–02 | 2.59e–02 | 1.29e–02 | 6.90e–03 | 3.60e–03 | 1.80e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.86 | 1.01 | 0.90 | 0.94 | 1.00 | Â | |
K | 4 | 4 | 3 | 3 | 3 | 3 | |
2−12 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.10e–02 | 2.83e–02 | 1.50e–02 | 7.80e–03 | 3.90e–03 | 2.00e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.85 | 0.92 | 0.94 | 1.00 | 0.96 | Â | |
K | 5 | 5 | 6 | 4 | 3 | 3 | |
2−14 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.54e–02 | 3.03e–02 | 1.62e–02 | 8.40e–03 | 4.30e–03 | 2.20e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.87 | 0.90 | 0.95 | 0.97 | 0.97 | Â | |
K | 6 | 5 | 4 | 4 | 4 | 3 | |
2−16 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.98e–02 | 3.14e–02 | 1.66e–02 | 8.60e–03 | 4.40e–03 | 2.30e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.93 | 0.92 | 0.95 | 0.97 | 0.94 | Â | |
K | 8 | 7 | 6 | 4 | 4 | 4 | |
2−18 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.89e–02 | 3.14e–02 | 1.68e–02 | 8.70e–03 | 4.50e–03 | 2.30e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.91 | 0.90 | 0.95 | 0.95 | 0.97 | Â | |
K | 8 | 6 | 5 | 5 | 4 | 4 | |
2−20 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.86e–02 | 3.14e–02 | 1.69e–02 | 8.70e–03 | 4.50e–03 | 2.30e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.90 | 0.89 | 0.96 | 0.95 | 0.97 | Â | |
K | 9 | 8 | 6 | 5 | 6 | 4 |