Remarkable upper bounds for the interpolation error constants on the triangles

TL;DR

This paper establishes sharp, simple-form upper bounds for interpolation error constants on triangles, crucial for finite element analysis, using numerical verification and asymptotic analysis, with broader applicability to norm inequalities.

Contribution

It introduces new sharp upper bounds for interpolation error constants on triangles, demonstrating the effectiveness of numerical verification methods in mathematical proofs.

Findings

Upper bounds are sharp and simple to compute

Numerical verification confirms the bounds

Asymptotic analysis supports the bounds' validity

Abstract

We introduce remarkable upper bounds for the interpolation error constants on triangles, which are sharp and given by simple formulas. These constants are crucial in analyzing interpolation errors, particularly those associated with the Finite Element Method. In this study, we proved boundness via the numerical verification method and asymptotic analysis. This study is also essential in that it demonstrates a valuable application of the numerical verification method. The proof process of this study may be applied to the proof of various other norm inequalities.