Table 3 Errors and order convergence for \(u_{\mathrm{ex}}^{1,\beta }\) using (42) with \(r= \frac{2-\beta }{\beta }\)

β = 0.2

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

1.992E−2

5.794E−3

1.670E−3

4.801E−4

1.379E−4

3.961E−5

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

1.782

1.795

1.798

1.800

1.800

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

5.719E−1

3.434E−1

1.894E−1

9.953E−2

5.104E−2

2.584E−2

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

0.736

0.859

0.928

0.964

0.982

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

0.921

0.981

0.994

1.004

1.005

β = 0.4

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

3.875E−2

1.307E−2

4.346E−3

1.437E−3

4.745E−4

1.566E−4

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

1.567

1.589

1.596

1.599

1.600

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

3.110E−1

1.691E−1

8.825E−2

4.511E−2

2.281E−2

1.147E−2

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

0.879

0.938

0.968

0.984

0.992

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

0.999

1.020

1.023

1.020

1.016

β = 0.6

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

7.709E−2

3.042E−2

1.171E−2

4.465E−3

1.696E−3

6.432E−4

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

1.341

1.377

1.391

1.397

1.399

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

1.927E−1

1.011E−1

5.184E−2

2.627E−2

1.323E−2

6.639E−3

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

0.930

0.963

0.981

0.990

0.995

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

1.024

1.037

1.038

1.034

1.030

β = 0.8

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

1.551E−1

7.262E−2

3.268E−2

1.444E−2

6.326E−3

2.761E−3

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

1.095

1.152

1.179

1.191

1.196

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

1.551E−1

7.262E−2

3.303E−2

1.668E−2

8.391E−3

4.210E−3

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

1.095

1.137

0.985

0.991

0.995

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

1.016

1.031

1.036

1.037

1.037