arXiv subjects

Jérôme Droniou

Publications and source records attributed to Jérôme Droniou.

At least 19 recordsLinked to original sources

arXiv2026

Design and homological analysis of twisted and BGG Stokes-de Rham complexes on polygonal meshes

We design a discrete Bernstein--Gelfand--Gelfand (BGG) diagram in a two-dimensional setting on polygonal meshes based on the DDR framework; the diagram is made of a discrete Stokes polygonal complex and a tensorised Discrete de Rham complex, and the BGG construction leads to novel twisted and Hessian complexes applicable on generic polygonal meshes; such complexes enable, in particular, discretisations of plate problems. Complete homological properties of the discrete Stokes complex and of the complexes built from the BGG diagram are established.

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arXiv2026

Analytical properties of polygonal Stokes and BGG Hessian complexes with application to Kirchhoff--Love plates

We develop and analyse a new arbitrary-order method for the Kirchhoff--Love plate problem on general polygonal meshes. The construction relies on a discrete Hessian complex obtained through the Bernstein--Gelfand--Gelfand construction, which stacks a Stokes complex on top of a tensorised de Rham complex. A key ingredient of the analysis is a detailed study of the former. We specifically establish primal and adjoint consistency estimates, as well as uniform Poincaré inequalities, and use these results to derive the analytical properties of the resulting discrete Hessian complex. The proposed method is then shown to be coercive and to converge with order $k+1$ with respect to the mesh size, where $k\ge 0$ is the polynomial degree of the complex. Numerical experiments on polygonal meshes confirm the theoretical convergence rates.

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arXiv2026

Ghost stabilisation for cut finite element exterior calculus

We introduce the cut finite element method in the language of finite element exterior calculus, by formulating a stabilisation -- for any form degree -- that makes the method robust with respect to the position of the interface relative to the mesh. We prove that the $L^2$-norm on the physical domain augmented with this stabilisation is uniformly equivalent to the $L^2$-norm on the ``active'' mesh that contains all the degrees of freedom of the finite element space (including those external to the physical domain). We show how this CutFEEC method can be applied to discretize the Hodge Laplace equations on an unfitted mesh, in any dimension and any topology, and prove stability and optimal convergence. Two numerical examples are provided, with convergence and condition number scaling independent of the position of the boundary with respect to the background mesh. We first solve the Hodge Laplace equation on a conforming finite element space of $H^{\text{curl}}$ posed on a filled torus. The second example extends beyond the Hodge Laplace problem and illustrates the importance of stabilisation for pressure robustness in unfitted schemes, using an $H^{\text{curl}}$-formulation for the Stokes equations.

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arXiv2026

A Hybrid High-Order method for the power-law Brinkman problem with robust error estimates in all regimes

In this work we propose and analyze a new Hybrid High-Order method for the Brinkman problem for fluids with power-law viscosity. The proposed method supports general meshes and arbitrary approximation orders and is robust in all regimes, from pure (power-law) Stokes to pure Darcy. Robustness is reflected by error estimates that distinguish the contributions from Stokes- and Darcy-dominated elements as identified by an appropriate dimensionless number, and that additionally account for pre-asymptotic orders of convergence. Theoretical results are illustrated by a complete panel of numerical experiments.

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arXiv2026

Key challenges and bridges among convergence analysis techniques for polytopal methods

Polytopal methods provide a flexible framework for the numerical approximation of partial differential equations on general meshes. Their convergence analysis raises specific challenges due to their inherently non-conforming nature and, in many cases, the fully discrete nature of their solution. Two main techniques are considered: the virtual-function approach, used, e.g., in the context of Virtual Element Methods, and the fully discrete approach, which underlies, e.g., the Discrete de Rham method. We introduce here a novel framework based on the notion of conforming liftings, namely bounded and consistent mappings from the discrete space into the continuous space. This approach bridges the virtual and fully discrete viewpoints, clarifies the role of norm equivalence for virtual functions, and leads to a decomposition of the consistency error usable for polytopal methods. The three approaches are demonstrated on a model problem, which provides the opportunity to discuss relevant technical points. Bridges with the convergence properties of discrete differential complexes are also built.

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arXiv2026

A low-order hybrid method for the variable-density incompressible Navier-Stokes equations

In this work we introduce and analyse a new low-order method for the variable-density incompressible Navier-Stokes equations. The main novelty of the proposed method lies in the support of general meshes, possibly including polygonal or polyhedral elements as well as non-matching interfaces. We carry out a complete analysis, showing stability, existence and uniqueness of a discrete solution, and convergence of the latter to a suitably defined weak solution of the continuous problem. Numerical tests validate the theoretical results.

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arXiv2025

A Reynolds-semi-robust method with hybrid velocity and pressure for the unsteady incompressible Navier--Stokes equations

In this paper we propose and analyze a new Finite Element method for the solution of the two- and three-dimensional incompressible Navier--Stokes equations based on a hybrid discretization of both the velocity and pressure variables. The proposed method is pressure-robust, i.e., irrotational forcing terms do not affect the approximation of the velocity, and Reynolds-quasi-robust, with error estimates that, for smooth enough exact solutions, do not depend on the inverse of the viscosity. We carry out an in-depth convergence analysis highlighting pre-asymptotic convergence rates and validate the theoretical findings with a complete set of numerical experiments.

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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.

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arXiv2025

Conforming lifting and adjoint consistency for the Discrete de Rham complex of differential forms

Discrete de Rham (DDR) methods provide non-conforming but compatible approximations of the continuous de Rham complex on general polytopal meshes. Owing to the non-conformity, several challenges arise in the analysis of these methods. In this work, we design conforming liftings on the DDR spaces, that are right-inverse of the interpolators and can be used to solve some of these challenges. We illustrate this by tackling the question of the global integration-by-part formula. By non-conformity of the discrete complex, this formula involves a residual -- which can be interpreted as a consistency error on the adjoint of the discrete exterior derivative -- on which we obtain, using the conforming lifting, an optimal bound in terms of the mesh size. Our analysis is carried out in the polytopal exterior calculus framework, which allows for unified proofs for all the spaces and operators in the DDR complex. Moreover, the liftings are explicitly constructed in finite element spaces on a simplicial submesh of the underlying polytopal mesh, which gives more control on the resulting functions (e.g., discrete trace and inverse inequalities).

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arXiv2025

Commuting quasi-interpolators and Maxwell compactness for a polytopal de Rham complex

We establish Maxwell compactness results for the Discrete De Rham (DDR) polytopal complex: sequences in this polytopal complex with bounded discrete $\boldsymbol{H}(\mathbf{curl})$ (resp. discrete $\boldsymbol{H}(\mathrm{div})$) norm and orthogonal to discrete gradients (resp. discrete curls) have $L^2$-relatively compact potential reconstructions. The proof of these results hinges on the design of novel quasi-interpolators, that map the minimal-regularity de Rham spaces onto the discrete DDR spaces and form a commuting diagram. A full set of (primal and adjoint) consistency properties is established for these quasi-interpolators, which paves the way to convergence proofs, under minimal-regularity assumptions, of DDR schemes for partial differential equations based on the de Rham complex. Our analysis is performed with generic mixed boundary conditions, also covering the cases of no boundary conditions or fully homogeneous boundary conditions, and leverages recently introduced liftings from the DDR complex to the continuous de Rham complex.

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arXiv2025

Efficient iterative linearised solvers for numerical approximations of stochastic Stefan problems

We present iterative solvers to approximate the solution of numerical schemes for stochastic Stefan problems. After briefly talking about the convergence results, we tackle the question of efficient strategies for solving the nonlinear equation associated with this scheme. We explore several approaches, from a standard Newton technique to linearised solvers. The latter offer the advantage of using the same coefficient matrix of the linearised system in each nonlinear iteration, for all time steps, and across all realisations of the Brownian motions. As a consequence, the system can be factorised once and for all. Although the linearised approach has a slower convergence rate, our sensitivity analysis and the use of adaptive tolerance in both deterministic and stochastic cases provide valuable insights for choosing the most effective solver across various scenarii.

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arXiv2025

From Finite Elements to Hybrid High-Order methods

This document contains lecture notes from the Ph.D. course given at Scuola Superiore Meridionale by Daniele Di Pietro in February 2025. The goal of the course is to provide an overview of polytopal methods, focusing on the Hybrid High-Order (HHO) method. As a starting point, we study the Crouzeix-Raviart method for a pure diffusion equation, with particular focus on its stability. We then show that, switching to a fully discrete point of view, it is possible to generalize it first to polyhedral meshes and then to arbitrary order, leading to a method that belongs to the HHO family. A study of the stability and consistency of this method reveals the need for a stabilization term, for which we identify two key properties.

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arXiv2025

A polytopal discrete de Rham complex on manifolds, with application to the Maxwell equations

We design in this work a discrete de Rham complex on manifolds. This complex, written in the framework of exterior calculus, has the same cohomology as the continuous de Rham complex, is of arbitrary order of accuracy and, in principle, can be designed on meshes made of generic elements (that is, elements whose boundary is the union of an arbitrary number of curved edges/faces). Notions of local (full and trimmed) polynomial spaces are developed, with compatibility requirements between polynomials on mesh entities of various dimensions. We give explicit constructions of such polynomials in 2D, for some meshes made of curved triangles or quadrangles (such meshes are easy to design in many cases, starting from a few charts describing the manifold). The discrete de Rham complex is then used to set up a scheme for the Maxwell equations on a 2D manifold without boundary, and we show that a natural discrete version of the constraint linking the electric field and the electric charge density is satisfied. Numerical examples are provided on the sphere and the torus, based on bespoke analytical solutions and meshes on each of these manifolds.

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arXiv2024

Arbitrary-order pressure-robust DDR and VEM methods for the Stokes problem on polyhedral meshes

This paper contains two major contributions. First we derive, following the discrete de Rham (DDR) and Virtual Element (VEM) paradigms, pressure-robust methods for the Stokes equations that support arbitrary orders and polyhedral meshes. Unlike other methods presented in the literature, pressure-robustness is achieved here without resorting to an $\boldsymbol{H}({\rm div})$-conforming construction on a submesh, but rather projecting the volumetric force onto the discrete $\boldsymbol{H}({\bf curl})$ space. The cancellation of the pressure error contribution stems from key commutation properties of the underlying DDR and VEM complexes. The pressure-robust error estimates in $h^{k+1}$ (with $h$ denoting the meshsize and $k\ge 0$ the polynomial degree of the DDR or VEM complex) are proven theoretically and supported by a panel of three-dimensional numerical tests. The second major contribution of the paper is an in-depth study of the relations between the DDR and VEM approaches. We show, in particular, that a complex developed following one paradigm admits a reformulation in the other, and that couples of related DDR and VEM complexes satisfy commuting diagram properties with the degrees of freedom maps.

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arXiv2024

Two arbitrary-order constraint-preserving schemes for the Yang--Mills equations on polyhedral meshes

Two numerical schemes are proposed and investigated for the Yang--Mills equations, which can be seen as a nonlinear generalisation of the Maxwell equations set on Lie algebra-valued functions, with similarities to certain formulations of General Relativity. Both schemes are built on the Discrete de Rham (DDR) method, and inherit from its main features: an arbitrary order of accuracy, and applicability to generic polyhedral meshes. They make use of the complex property of the DDR, together with a Lagrange-multiplier approach, to preserve, at the discrete level, a nonlinear constraint associated with the Yang--Mills equations. We also show that the schemes satisfy a discrete energy dissipation (the dissipation coming solely from the implicit time stepping). Issues around the practical implementations of the schemes are discussed; in particular, the assembly of the local contributions in a way that minimises the price we pay in dealing with nonlinear terms, in conjunction with the tensorisation coming from the Lie algebra. Numerical tests are provided using a manufactured solution, and show that both schemes display a convergence in $L^2$-norm of the potential and electrical fields in $\mathcal O(h^{k+1})$ (provided that the time step is of that order), where $k$ is the polynomial degree chosen for the DDR complex. We also numerically demonstrate the preservation of the constraint.

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arXiv2024

Numerical analysis of a mixed-dimensional poromechanical model with frictionless contact at matrix-fracture interfaces

We present a complete numerical analysis for a general discretization of a coupled flow-mechanics model in fractured porous media, considering single-phase flows and including frictionless contact at matrix-fracture interfaces, as well as nonlinear poromechanical coupling. Fractures are described as planar surfaces, yielding the so-called mixed- or hybrid-dimensional models. Small displacements and a linear elastic behavior are considered for the matrix. The model accounts for discontinuous fluid pressures at matrix-fracture interfaces in order to cover a wide range of normal fracture conductivities. The numerical analysis is carried out in the Gradient Discretization framework, encompassing a large family of conforming and nonconforming discretizations. The convergence result also yields, as a by-product, the existence of a weak solution to the continuous model. A numerical experiment in 2D is presented to support the obtained result, employing a Hybrid Finite Volume scheme for the flow and second-order finite elements ($\mathbb P_2$) for the mechanical displacement coupled with face-wise constant ($\mathbb P_0$) Lagrange multipliers on fractures, representing normal stresses, to discretize the contact conditions.

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arXiv2024

Analysis of a VEM-fully discrete polytopal scheme with bubble stabilisation for contact mechanics with Tresca friction

This work performs the convergence analysis of the polytopal nodal discretisation of contact-mechanics (with Tresca friction) recently introduced in [18] in the framework of poro-elastic models in fractured porous media. The scheme is based on a mixed formulation, using face-wise constant approximations of the Lagrange multipliers along the fracture network and a fully discrete first order nodal approximation of the displacement field. The displacement field is enriched with additional bubble degrees of freedom along the fractures to ensure the inf-sup stability with the Lagrange multiplier space. It is presented in a fully discrete formulation, which makes its study more straightforward, but also has a Virtual Element interpretation. The analysis establishes an abstract error estimate accounting for the fully discrete framework and the non-conformity of the discretisation. A first order error estimate is deduced for sufficiently smooth solutions both for the gradient of the displacement field and the Lagrange multiplier. A key difficulty of the numerical analysis is the proof of a discrete inf-sup condition, which is based on a non-standard $H^{-1/2}$-norm (to deal with fracture networks) and involves the jump of the displacements, not their traces. The analysis also requires the proof of a discrete Korn inequality for the discrete displacement field which takes into account fracture networks. Numerical experiments based on analytical solutions confirm our theoretical findings

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arXiv2024

A bubble VEM-fully discrete polytopal scheme for mixed-dimensional poromechanics with frictional contact at matrix fracture interfaces

The objective of this article is to address the discretisation of fractured/faulted poromechanical models using 3D polyhedral meshes in order to cope with the geometrical complexity of faulted geological models. A polytopal scheme is proposed for contact-mechanics, based on a mixed formulation combining a fully discrete space and suitable reconstruction operators for the displacement field with a face-wise constant approximation of the Lagrange multiplier accounting for the surface tractions along the fracture/fault network. To ensure the inf--sup stability of the mixed formulation, a bubble-like degree of freedom is included in the discrete space of displacements (and taken into account in the reconstruction operators). It is proved that this fully discrete scheme for the displacement is equivalent to a low-order Virtual Element scheme, with a bubble enrichment of the VEM space. This $\mathbb{P}^1$-bubble VEM--$\mathbb{P}^0$ mixed discretization is combined with an Hybrid Finite Volume scheme for the Darcy flow. All together, the proposed approach is adapted to complex geometry accounting for network of planar faults/fractures including corners, tips and intersections; it leads to efficient semi-smooth Newton solvers for the contact-mechanics and preserve the dissipative properties of the fully coupled model. Our approach is investigated in terms of convergence and robustness on several 2D and 3D test cases using either analytical or numerical reference solutions both for the stand alone static contact mechanical model and the fully coupled poromechanical model.

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