Correction to: Japan Journal of Industrial and Applied Mathematics (2021) 38:163–191

This article is a correction to our previous paper [1]. The geometric parameters \(\alpha _{\max }\) and \(\alpha _{\min }\), which represent the maximum and minimum edge lengths of the simplex under consideration, play a significant role in [1]. Theorems 2 and 3 claim that the interpolation errors can be estimated without the ratio \(\alpha _{\max }/\alpha _{\min }\). Unfortunately, the proofs of Theorems 2 and 3 contain some mistakes. The ratio \(\alpha _{\max }/\alpha _{\min }\) cannot be removed from by the technique proposed in [1].

As a correction to Theorem 2, Theorems A and B are given in [3]. Theorem A presents an error estimate with the ratio \(\alpha _{\max }/\alpha _{\min }\) under the assumptions of Theorem 2 in [1]. Theorem B presents an error estimate without the ratio \(\alpha _{\max }/\alpha _{\min }\) under stronger assumptions than in Theorem 2.

Using a new approach, we have succeeded in completing the proof of Theorem 3 in [1]. See Theorems 2 and 3 in [2] for the detail.