Theory and Modern Applications
Table 1 Numerical results with \(\varepsilon _{1}=2^{-9}\) for Example 1
\(\varepsilon _{2}\) |  | N = 32 | N = 64 | N = 128 | N = 256 | N = 512 | N = 1024 |
|---|---|---|---|---|---|---|---|
20 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 4.18e–02 | 2.31e–02 | 1.25e–02 | 6.40e–03 | 3.30e–03 | 1.70e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.86 | 0.89 | 0.97 | 0.96 | 0.96 | Â | |
K | 5 | 4 | 4 | 3 | 3 | 3 | |
2−2 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 4.62e–02 | 2.65e–02 | 1.42e–02 | 7.40e–03 | 3.80e–03 | 2.00e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.80 | 0.90 | 0.94 | 0.96 | 0.93 | Â | |
K | 5 | 4 | 4 | 3 | 3 | 3 | |
2−4 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 4.66e–02 | 2.68e–02 | 1.44e–02 | 7.30e–03 | 3.80e–03 | 1.90e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.80 | 0.90 | 0.98 | 0.94 | 1.00 | Â | |
K | 5 | 4 | 4 | 3 | 3 | 3 | |
2−6 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.11e–02 | 2.73e–02 | 1.42e–02 | 7.30e–03 | 3.70e–03 | 1.90e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.90 | 0.94 | 0.96 | 0.98 | 0.96 | Â | |
K | 5 | 4 | 4 | 3 | 3 | 3 | |
2−8 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 4.80e–02 | 2.59e–02 | 1.32e–02 | 6.90e–03 | 3.50e–03 | 1.80e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.89 | 0.97 | 0.94 | 0.98 | 0.96 | Â | |
K | 4 | 5 | 3 | 3 | 3 | 3 | |
2−10 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 4.95e–02 | 2.62e–02 | 1.36e–02 | 6.80e–03 | 3.50e–03 | 1.80e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.92 | 0.95 | 1.00 | 0.96 | 0.96 | Â | |
K | 8 | 4 | 4 | 3 | 3 | 3 | |
2−12 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.35e–02 | 2.93e–02 | 1.53e–02 | 7.90e–03 | 3.90e–03 | 2.00e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.87 | 0.94 | 0.95 | 1.02 | 0.96 | Â | |
K | 5 | 5 | 5 | 4 | 3 | 3 | |
2−14 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.49e–02 | 3.06e–02 | 1.64e–02 | 8.40e–03 | 4.40e–03 | 2.10e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.84 | 0.90 | 0.97 | 0.93 | 1.07 | Â | |
K | 7 | 6 | 5 | 4 | 4 | 3 | |
2−16 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.78e–02 | 3.06e–02 | 1.67e–02 | 8.70e–03 | 4.40e–03 | 2.30e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.92 | 0.87 | 0.94 | 0.98 | 0.94 | Â | |
K | 6 | 5 | 6 | 5 | 4 | 4 | |
2−18 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.78e–02 | 3.25e–02 | 1.66e–02 | 8.40e–03 | 4.40e–03 | 2.30e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.83 | 0.97 | 0.98 | 0.93 | 0.94 | Â | |
K | 7 | 8 | 5 | 4 | 4 | 4 | |
2−20 | \(E_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 5.78e–02 | 3.09e–02 | 1.72e–02 | 8.70e–03 | 4.30e–03 | 2.30e–03 |
\(r_{\varepsilon _{1},\varepsilon _{2}}^{N}\) | 0.90 | 0.85 | 0.98 | 1.02 | 0.90 | Â | |
K | 7 | 6 | 9 | 5 | 4 | 4 |