# Lectures on error analysis of interpolation on simplicial triangulations without the shape-regularity assumption Part 1: Lagrange interpolation on triangles

## TL;DR

This paper reviews error analysis of finite element interpolation on triangles without assuming shape regularity, emphasizing the role of circumradius, aimed at researchers and students.

## Contribution

It provides a clear explanation of error estimates for finite element methods on non-shape regular triangulations, focusing on circumradius importance.

## Key findings

- Error estimates can be established without shape regularity assumptions.
- Circumradius is a key factor in error analysis.
- The paper aims to serve as educational material for researchers and students.

## Abstract

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 regularity assumption may appear when we use adaptive mesh refinement. In this manuscript, we attempt to establish an error analysis approach without the shape regularity assumption on triangulations. We have presented several papers on the error analysis of finite element methods on non-shape regular triangulations. The main points in these papers are that, in the error estimates of finite element methods, the circumradius of the triangles is one of the most important factors. The purpose of this manuscript is to provide a simple and plain explanation of the results to researchers and, in particular, graduate students who are interested in the subject. Therefore, this 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.