Abstract
AbstractIt is known that piecewise linear interpolation of functions of one variable is uniformly bounded with an H1‐stability constant of one. In [1], we considered the nodal interpolation operator acting between spaces of piecewise linear functions and presented an elementary proof by minimizing a functional representing the H1‐semi‐norm. In this note, a standard approximation argument is applied generalizing the result to piecewise linear interpolation of all functions in W1,p on a real interval, 1 ≤ p ≤ ∞. We also comment on alternative proofs and finally give a counterexample for piecewise linear interpolation in 2D. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)
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