A Rocq Formalization of Simplicial Lagrange Finite Elements

2026-04-22

AI Summary

This study addresses the lack of a rigorous formal definition of "finite elements" in the finite element method by proposing a formal framework within the Rocq proof assistant based on record types, wherein a finite element is modeled as a structure comprising geometric data and validity proofs. The work presents the first complete formalization of simplicial Lagrange finite elements of arbitrary dimension and polynomial degree in a proof assistant, and rigorously verifies their unisolvence property using foundational theories of finite families, affine spaces, and multivariate polynomials. This achievement yields a general definition and correctness proof for simplicial Lagrange finite elements with uniform nodal distributions, thereby establishing a formal foundation for the verification of scientific computing software.

Technology Category

• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Web Mining and Content Analysis: Web data provenance, reliability, and authenticity
• Systems and Infrastructure for Web, Mobile and WoT: Virtualization and resource management in Web systems and infrastructures
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

Formalization of mathematics is a major topic, that includes in particular numerical analysis, towards proofs of scientific computing programs. The present study is about the finite element method, a popular method to numerically solve partial differential equations. In the long-term goal of proving its correctness, we focus here on the formal definition of what is a finite element. Mathematically, a finite element describes what happens in a cell of a mesh. It notably includes the geometry of the cell, the polynomial approximation space, and a finite set of linear forms that computationally characterizes the polynomials. Formally, we design a finite element as a record in the Rocq proof assistant with both values (such as the vertices of the cell) and proofs of validity (such as the dimension of the approximation space). The decisive validity proof is unisolvence, that makes the previous characterization unique. We then instantiate this record with the most popular and useful, the simplicial Lagrange finite elements for evenly distributed nodes, for any dimension and any polynomial degree, including the difficult unisolvence proof. These proofs require many results (definitions, lemmas, canonical structures) about finite families, affine spaces, multivariate polynomials, in the context of finite or infinite-dimensional spaces.

Problem

• finite element
• formalization
• Lagrange finite elements
• unisolvence
• numerical analysis

Innovation

• formal verification
• finite element method
• simplicial Lagrange elements
• unisolvence
• proof assistant