Theory and Modern Applications

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

$$r = \frac{2-\beta }{\beta }$$ n = 32 n = 64 n = 128 n = 256 n = 512 n = 1028
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