Table 1 Errors and order convergence for \(u_{\mathrm{ex}}^{1,\beta }\) using (41) with \(r= \frac{2-\beta }{\beta }\)

β = 0.2

\(E_{\mathrm{max}_{I}}^{n}\)

2.479E−2

7.233E−3

2.087E−3

6.000E−4

1.724E−4

4.951E−5

\(\text{rate}_{{E_{\mathrm{max}_{I}}^{n}}}\)

1.777

1.793

1.798

1.799

1.800

\(E_{\mathrm{max}}^{n}\)

5.647E−1

3.356E−1

1.823E−1

9.430E−2

4.748E−2

2.354E−2

\(\text{rate}_{E_{\mathrm{max}}^{n}}\)

0.751

0.880

0.951

0.990

1.012

\(\text{rate}_{E_{\mathrm{conv}}^{n}}\)

0.700

0.746

0.767

0.777

0.783

β = 0.4

\(E_{\mathrm{max}_{I}}^{n}\)

6.318E−2

2.163E−2

7.225E−3

2.393E−3

7.906E−4

2.609E−4

\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\)

1.547

1.582

1.594

1.598

1.599

\(E_{\mathrm{max}}^{n}\)

2.863E−1

1.468E−1

7.060E−2

3.209E−2

1.358E−2

5.086E−3

\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\)

0.964

1.056

1.138

1.240

1.417

\(\text{rate}_{E_{\mathrm{conv}}^{n}}\)

0.565

0.581

0.589

0.593

0.596

β = 0.6

\(E_{\mathrm{max}_{I}}^{n}\)

1.758E−1

7.328E−2

2.886E−2

1.110E−2

4.231E−3

1.607E−3

\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\)

1.263

1.344

1.378

1.392

1.397

\(E_{\mathrm{max}}^{n}\)

1.758E−1

7.328E−2

2.886E−2

1.110E−2

1.225E−2

1.296E−2

\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\)

1.263

1.344

1.378

-0.142

-0.082

\(\text{rate}_{E_{\mathrm{conv}}^{n}}\)

0.398

0.405

0.407

0.407

0.406

β = 0.8

\(E_{\mathrm{max}_{I}}^{n}\)

5.185E−1

2.947E−1

1.479E−1

6.900E−2

3.100E−2

1.369E−2

\(\text{rate}_{E_{\mathrm{max}_{I}}^{n}}\)

0.815

0.994

1.100

1.154

1.180

\(E_{\mathrm{max}}^{n}\)

5.185E−1

2.947E−1

1.479E−1

7.440E−2

7.131E−2

6.516E−2

\(\text{rate}_{{E_{\mathrm{max}}^{n}}}\)

0.815

0.994

0.992

0.061

0.130

\(\text{rate}_{E_{\mathrm{conv}}^{n}}\)

0.214

0.217

0.218

0.216

0.215