On the circumradius condition for piecewise linear triangular elements
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We discuss the error analysis of linear interpolation on triangular elements. We claim that the circumradius condition is more essential than the well-known maximum angle condition for convergence of the finite element method, especially for the linear Lagrange finite element. Numerical experiments show that this condition is the best possible. We also point out that the circumradius condition is closely related to the definition of surface area.
KeywordsLinear interpolation The circumradius condition The finite element method Schwarz’s example The definition of surface area
Mathematics Subject Classification65D05 65N30 26B15
The first author is supported by Inamori Foundation and JSPS Grant-in-Aid for Young Scientists (B) 22740059. The second author is partially supported by JSPS Grant-in-Aid for Scientific Research (C) 22440139 and Grant-in-Aid for Scientific Research (B) 23340023. The authors thank the anonymous referee for valuable comments and for drawing the authors’ attention to Besicovitch’s paper .
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