We present a general theory of interpolation error estimates for smooth functions and inverse inequalities on anisotropic meshes. In our theory, the error of interpolation is bound in terms of the diameter of a simplex and a geometric parameter. In the two-dimensional case, our geometric parameter is equivalent to the circumradius of a triangle. In the three-dimensional case, our geometric parameter also represents the flatness of a tetrahedron. This paper also includes corrections to an error in "General theory of interpolation error estimates on anisotropic meshes" (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191), in which Theorem 2 was incorrect.
翻译：我们提出了一个关于光滑函数的内推误估计总理论和对厌异色粒的反不平等性理论。 在我们的理论中,内推误以简单x和几何参数的直径为界。 在二维的案例中,我们的几何参数相当于三角形的环形。在三维的案例中,我们的几何参数也代表四面体的平面。本文还包括对“对厌异色粒粒子的内推误估计一般理论”中错误的更正(日本工业和应用数学杂志,38 (2021) 163-191), 其中Theorem 2不正确。
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