Error constant estimation under the maximum norm for linear Lagrange interpolation

TL;DR

This paper introduces an efficient FEM-based algorithm to rigorously estimate the maximum norm error constant for linear Lagrange interpolation over triangular domains, addressing the challenge of the maximum norm constraint.

Contribution

It presents a novel method combining orthogonality and Bernstein representation properties to evaluate the interpolation error constant under the maximum norm.

Findings

Numerical bounds for error constants on various triangles.

The proposed method effectively computes rigorous error bounds.

Verification of the method's efficiency through numerical experiments.

Abstract

For the Lagrange interpolation over a triangular domain, we propose an efficient algorithm to rigorously evaluate the interpolation error constant under the maximum norm by using the finite element method (FEM). In solving the optimization problem corresponding to the interpolation error constant, the maximum norm in the constraint condition is the most difficult part to process. To handle this difficulty, a novel method is proposed by combining the orthogonality of the interpolation associated to the Fujino--Morley FEM space and the convex-hull property of the Bernstein representation of functions in the FEM space. Numerical results for the lower and upper bounds of the interpolation error constant for triangles of various types are presented to verify the efficiency of the proposed method.