Explicit Bound for Quadratic Lagrange Interpolation Constant on Triangular Finite Elements

TL;DR

This paper develops an explicit, verified bound for the quadratic Lagrange interpolation error constant on triangles, using eigenvalue problems and numerical methods to ensure accuracy across various triangle shapes.

Contribution

It introduces an algorithm to compute explicit bounds for the interpolation constant, combining eigenvalue analysis with numerical verification for improved accuracy.

Findings

Upper bounds obtained via eigenvalue problems

Lower bounds derived from Rayleigh-Ritz method

Numerical results demonstrate bounds' sharpness across triangle shapes

Abstract

For the quadratic Lagrange interpolation function, an algorithm is proposed to provide explicit and verified bound for the interpolation error constant that appears in the interpolation error estimation. The upper bound for the interpolation constant is obtained by solving an eigenvalue problem along with explicit lower bound for its eigenvalues. The lower bound for interpolation constant can be easily obtained by applying the Rayleigh-Ritz method. Numerical computation is performed to demonstrate the sharpness of lower and upper bounds of the interpolation constants over triangles of different shapes. An online computing demo is available at http://www.xfliu.org/onlinelab/.