General theory of interpolation error estimates on anisotropic meshes

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We propose a general theory of estimating interpolation error for smooth functions in two and three dimensions. 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. Through the introduction of the geometric parameter, the error estimates newly obtained can be applied to cases that violate the maximum-angle condition.

29 pages, 2 figures. In "General theory of interpolation error estimates on anisotropic meshes" (Japan Journal of Industrial and Applied Mathematics, 38 (2021) 163-191), Theorem 2 has been found to be incorrect and misleading. Corrections to an error are given in "General theory of interpolation error estimates on anisotropic meshes, part II", arXiv:2106.03339

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