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This is the second lecture note on the error analysis of interpolation on simplicial elements without the shape regularity assumption (the previous one is arXiv:1908.03894). In this manuscript, we explain the error analysis of Lagrange interpolation on (possibly anisotropic) tetrahedrons. The manuscript is not intended to be a research paper. We hope that, in the future, it will be merged into a textbook on the mathematical theory of the finite element methods.
In the error analysis of finite element methods, the shape regularity assumption on triangulations is typically imposed to obtain a priori error estimations. In practical computations, however, very thin or degenerated elements that violate the shape
This paper describes the analysis of Lagrange interpolation errors on tetrahedrons. In many textbooks, the error analysis of Lagrange interpolation is conducted under geometric assumptions such as shape regularity or the (generalized) maximum angle c
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 t
We present the error analysis of Lagrange interpolation on triangles. A new textit{a priori} error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on triangles are i
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