Lectures on error analysis of interpolation on simplicial triangulations without the shape-regularity assumption, Part 2: Lagrange interpolation on tetrahedrons

TL;DR

This paper discusses the error analysis of Lagrange interpolation on tetrahedral elements without assuming shape regularity, focusing on anisotropic cases, aiming to contribute to finite element method theory.

Contribution

It provides an in-depth explanation of error analysis for Lagrange interpolation on possibly anisotropic tetrahedrons without shape regularity assumptions.

Findings

Error bounds for Lagrange interpolation on anisotropic tetrahedrons

Extension of interpolation error analysis beyond shape-regular meshes

Foundation for future integration into finite element textbooks

Abstract

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.