arXiv subjects

Marien-Lorenzo Hanot

Publications and source records attributed to Marien-Lorenzo Hanot.

6 recordsLinked to original sources

arXiv2025

Uniform Poincaré inequalities for the discrete de Rham complex of differential forms

In this paper we prove discrete Poincaré inequalities that are uniform in the mesh size for the discrete de Rham complex of differential forms developed in [Bonaldi, Di Pietro, Droniou, and Hu, An exterior calculus framework for polytopal methods, J. Eur. Math. Soc., to appear, arXiv preprint 2303.11093]. We unify the underlying ideas behind the Poincaré inequalities for all differential operators in the sequence, extending the known inequalities for the gradient, curl, and divergence in three-dimensions to polytopal domains of arbitrary dimension and general topology. A key step in the proof involves deriving specific Poincaré inequalities for the cochain complex supported on the polytopal mesh. These inequalities are of independent interest, as they are useful, for instance, in establishing the existence and stability, on domains of generic topology, of solutions of schemes based on Mimetic Finite Differences, Compatible Discrete Operators or Discrete Geometric Approach.

math.NA↗

arXiv2025

Hodge-Dirac wave systems and structure-preserving discretizations of the linearized Einstein equations

We derive a reformulation of the linearized Arnowitt-Deser-Misner (ADM) equations as a Hodge-Dirac wave system with the divdiv complex, addressing challenges in numerical relativity such as gauge fixing, constraint propagation, and tensor symmetries. The differential and algebraic structures of the divdiv complex ensure the well-posedness of the formulation and facilitate structure-preserving discretization via finite element exterior calculus. We establish the well-posedness of this Hodge-Dirac wave equation and develop a discretization scheme applicable to both conforming and non-conforming discrete complexes, deriving error estimates under minimal assumptions.

gr-qc↗

arXiv2023

Uniform Poincaré inequalities for the Discrete de Rham complex on general domains

In this paper we prove Poincaré inequalities for the Discrete de Rham (DDR) sequence on a general connected polyhedral domain $Ω$ of $\mathbb{R}^3$. We unify the ideas behind the inequalities for all three operators in the sequence, deriving new proofs for the Poincaré inequalities for the gradient and the divergence, and extending the available Poincaré inequality for the curl to domains with arbitrary second Betti numbers. A key preliminary step consists in deriving "mimetic" Poincaré inequalities giving the existence and stability of the solutions to topological balance problems useful in general discrete geometric settings. As an example of application, we study the stability of a novel DDR scheme for the magnetostatics problem on domains with general topology.

math.NA↗

arXiv2023

A discrete three-dimensional divdiv complex on polyhedral meshes with application to a mixed formulation of the biharmonic problem

In this work, following the Discrete de Rham (DDR) paradigm, we develop an arbitrary-order discrete divdiv complex on general polyhedral meshes. The construction rests 1) on discrete spaces that are spanned by vectors of polynomials whose components are attached to mesh entities and 2) on discrete operators obtained mimicking integration by parts formulas. We provide an in-depth study of the algebraic properties of the local complex, showing that it is exact on mesh elements with trivial topology. The new DDR complex is used to design a numerical scheme for the approximation of biharmonic problems, for which we provide detailed stability and convergence analyses.

math.NA↗

arXiv2022

An arbitrary-order fully discrete Stokes complex on general polyhedral meshes

In this paper we present an arbitrary-order fully discrete Stokes complex on general polyhedral meshes. We enriche the fully discrete de Rham complex with the addition of a full gradient operator defined on vector fields and fitting into the complex. We show a complete set of results on the novelties of this complex: exactness properties, uniform Poincaré inequalities and primal and adjoint consistency. The Stokes complex is especially well suited for problem involving Jacobian, divergence and curl, like the Stokes problem or magnetohydrodynamic systems. The framework developed here eases the design and analysis of scheme for such problems. Schemes built that way are nonconforming and benefit from the exactness of the complex. We illustrate with the design and study of a scheme to solve the Stokes equations and validate the convergence rates with various numerical tests.

math.NA↗

arXiv2022

Numerical solution of the div-curl problem by finite element exterior calculus

We are interested in the numerical reconstruction of a vector field with prescribed divergence and curl in a general domain of R 3 or R 2 , not necessarily contractible. To this aim, we introduce some basic concepts of finite element exterieur calculus and rely heavily on recent results of P. Leopardi and A. Stern. The goal of the paper is to take advantage of the links between usual vector calculus and exterior calculus and show the interest of the exterior calculus framework, without too much prior knowledge of the subject. We start by describing the method used for contractible domains and its implementation using the FEniCS library (see fenicsproject.org). We then address the problems encountered with non contractible domains and general boundary conditions and explain how to adapt the method to handle these cases. Finally we give some numerical results obtained with this method, in dimension 2 and 3.

math.NA↗