Lectures on error analysis of interpolation on simplicial triangulations without the shape-regularity assumption and its applications to finite element methods, Part 1: two-dim
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In the error analysis of finite element methods, the shape-regularity assumption on triangulations is usually imposed to obtain anticipated error estimations. In practical computations, however, very "thin" or "degenerated" elements may appear, when we use adaptive mesh refinement. In this manuscript, we will try to establish an error analysis without the shape-regularity assumption on triangulations. The authors have presented several papers on error analysis of finite eleemnt methods on non-regular triangulation. The main points of those 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, to graduate students who are interested in the subject. Therefore, the manuscript is not intended as a research paper. The authors hope that it will be merged into a textbook on the mathematical theory of the finite element methods in future.
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