Table 1 Comparing errors with different types of norms and quasi-norms for Example 1 with \(p = 4.0\) and \(s = 0.6\) for SMRPI and EFG methods

\(|u-u_{h}|_{1,p}\)

EFG

4.024 × 10⁻³

1.740 × 10⁻³

8.403 × 10⁻⁴

4.116 × 10⁻⁴

2.049 × 10⁻⁴

SMRPI

1.334 × 10⁻¹

9.632 × 10⁻⁴

8.286 × 10⁻⁶

4.583 × 10⁻⁷

2.189 × 10⁻⁸

\(|u-u_{h}|_{u,p}\)

EFG

3.758 × 10⁻³

1.657 × 10⁻³

7.986 × 10⁻⁴

3.924 × 10⁻⁴

1.954 × 10⁻⁴

SMRPI

1.004 × 10⁻¹

6.520 × 10⁻⁴

7.601 × 10⁻⁶

4.113 × 10⁻⁷

1.398 × 10⁻⁷

\(|\sigma -\sigma _{h}|_{0,q}\)

EFG

7.751 × 10⁻³

3.239 × 10⁻³

1.559 × 10⁻³

7.678 × 10⁻⁴

3.831 × 10⁻⁴

SMRPI

3.335 × 10⁻¹

3.212 × 10⁻³

7.256 × 10⁻⁶

5.813 × 10⁻⁶

3.459 × 10⁻⁶

\(|\sigma -\sigma _{h}|_{\sigma ,q}\)

EFG

8.527 × 10⁻³

3.573 × 10⁻³

1.715 × 10⁻³

8.422 × 10⁻⁴

4.197 × 10⁻⁴

SMRPI

3.884 × 10⁻¹

7.932 × 10⁻³

9.886 × 10⁻⁵

5.983 × 10⁻⁶

3.289 × 10⁻⁷