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

From: Existence of a unique weak solution to a non-autonomous time-fractional diffusion equation with space-dependent variable order

\(r = \frac{2-\beta }{\beta }\) n = 32 n = 64 n = 128 n = 256 n = 512 n = 1028
β = 0.2 \(E_{\mathrm{max}_{I}}^{n}\) 1.733E−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