Table 4 Errors and order convergence for \(u_{\mathrm{ex}}^{2,\beta }\) using (42) with \(r= \frac{2-\beta }{\beta }\)

β = 0.2

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

1.733E−2

5.046E−3

1.455E−3

4.183E−4

1.202E−4

3.451E−5

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

1.780

1.794

1.798

1.800

1.800

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

5.105E−2

2.580E−2

1.279E−2

6.399E−3

3.188E−3

1.598E−3

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

0.984

1.012

1.000

1.005

0.996

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

1.083

1.058

1.004

1.039

1.052

β = 0.4

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

3.239E−2

1.096E−2

3.645E−3

1.206E−3

3.982E−4

1.314E−4

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

1.564

1.588

1.596

1.599

1.600

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

4.603E−2

2.310E−2

1.157E−2

5.786E−3

2.892E−3

1.444E−3

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

0.995

0.998

0.999

1.001

1.002

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

1.060

1.043

1.030

1.022

1.018

β = 0.6

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

6.202E−2

2.460E−2

9.487E−3

3.619E−3

1.375E−3

5.216E−4

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

1.334

1.374

1.390

1.396

1.399

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

6.202E−2

2.460E−2

1.028E−2

5.141E−3

2.568E−3

1.282E−3

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

1.334

1.258

1.000

1.001

1.003

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

1.059

1.051

1.043

1.036

1.031

β = 0.8

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

1.201E−1

5.673E−2

2.563E−2

1.134E−2

4.974E−3

2.172E−3

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

1.082

1.146

1.176

1.189

1.195

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

1.201E−1

5.673E−2

2.563E−2

1.134E−2

4.974E−3

2.172E−3

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

1.082

1.146

1.176

1.189

1.195

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

1.384

1.444

1.256

1.037

1.038