  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.
