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

β = 0.2

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

2.204E−2

6.439E−3

1.858E−3

5.344E−4

1.535E−4

4.410E−5

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

1.775

1.793

1.798

1.799

1.800

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

4.902E−2

2.435E−2

1.203E−2

5.917E−3

2.899E−3

1.413E−3

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

1.009

1.017

1.024

1.030

1.036

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

0.638

0.659

0.681

0.700

0.716

β = 0.4

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

5.527E−2

1.897E−2

6.342E−3

2.101E−3

6.943E−4

2.291E−4

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

1.543

1.581

1.594

1.598

1.599

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

5.527E−2

1.897E−2

8.380E−3

3.611E−3

1.421E−3

1.058E−3

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

1.543

1.179

1.214

1.346

0.426

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

0.523

0.539

0.553

0.564

0.573

β = 0.6

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

1.515E−1

6.342E−2

2.502E−2

9.630E−3

3.671E−3

1.394E−3

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

1.256

1.342

1.377

1.391

1.397

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

1.515E−1

6.342E−2

2.502E−2

9.630E−3

6.739E−3

5.411E−3

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

1.256

1.342

1.377

0.515

0.317

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

0.375

0.383

0.389

0.393

0.395

β = 0.8

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

4.380E−1

2.510E−1

1.265E−1

5.908E−2

2.656E−2

1.173E−2

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

0.803

0.989

1.098

1.153

1.179

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

4.380E−1

2.510E−1

1.265E−1

5.908E−2

2.697E−2

2.394E−2

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

0.803

0.989

1.098

1.131

0.172

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

0.199

0.202

0.203

0.204

0.204