The Hodge-Laplacian on the \v{C}ech-de Rham complex governs coupled problems

TL;DR

This paper develops a Hilbert complex framework for the cech-de Rham complex, enabling well-posed Hodge-Laplace problems that model various coupled physical systems such as flow and elasticity.

Contribution

It introduces a Hilbert space structure on the cech-de Rham complex, establishing a foundation for analyzing coupled problems via Hodge-Laplace equations.

Findings

Hodge-Laplace equations govern coupled physical systems

Framework ensures well-posedness of these problems

Applicable to systems like flow and elasticity

Abstract

By endowing the \v{C}ech-de Rham complex with a Hilbert space structure, we obtain a Hilbert complex with sufficient properties to allow for well-posed Hodge-Laplace problems. We observe that these Hodge-Laplace equations govern a class of coupled problems arising from physical systems including elastically attached strings, multiple-porosity flow systems and 3D-1D coupled flow models.