Lectures on the Error Analysis of Interpolation on Simplicial Triangulations without the Shape Regularity Assumption and Its Applications to Finite Element Methods Part 1: Lagrange Interpolation on Triangles

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 survey, 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 survey 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 survey is not intended to be a research paper. We hope that, in the near future, it will be merged into a textbook on the mathematical theory of the finite element methods.