Finite element method. Detailed proofs to be formalized in Coq

TL;DR

This paper provides detailed formal proofs for the construction of Lagrange finite elements of any degree on simplices, aiming to enhance the rigor and confidence in finite element method implementations.

Contribution

It offers comprehensive pen-and-paper proofs for finite element construction, facilitating formal verification in proof assistants like Coq.

Findings

Formal proofs for finite element construction provided

Enhances confidence in numerical PDE solutions

Supports formal verification in proof assistants

Abstract

To obtain the highest confidence on the correction of numerical simulation programs for the resolution of Partial Differential Equations (PDEs), one has to formalize the mathematical notions and results that allow to establish the soundness of the approach. The finite element method is one of the popular tools for the numerical resolution of a wide range of PDEs. The purpose of this document is to provide the formal proof community with very detailed pen-and-paper proofs for the construction of the Lagrange finite elements of any degree on simplices in positive dimension.