arXiv subjects

Daniele A. Di Pietro

Publications and source records attributed to Daniele A. Di Pietro.

At least 19 recordsLinked to original sources

arXiv2026

Numerical simulation of biological capsules based on interface-fitted polygonal meshes

In this work we introduce a numerical framework for the simulation of biological capsules based on meshes that adapt iteration-wise to the shape of the membrane. The discretization leverages a Hybrid High Order method (HHO) to solve a Stokes problem with a source term distributed on the interface between the internal and the external phases. The HHO discretization supports meshes composed of general polygons, permitting to use fitted meshes that can be easily generated by cutting the elements intersected by the interface. Moreover, polygonal meshes allow arbitrarily accurate interface discretizations without introducing bulk degrees of freedom. The presentation of the simulation algorithm is completed with a proof-of-concept numerical test on a well-established benchmark: the deformation of an elastic capsule immersed in a shear flow.

math.NA↗

arXiv2026

poly_rust: a domain-specific language for polytopal methods with a Rust interpreter

Background: Over the last two decades, considerable effort in the applied mathematics community has been devoted to the development of polytopal methods, i.e., numerical methods for partial differential equations that support meshes more general than those used by traditional finite element methods. Despite their flexibility and attractive mathematical properties, polytopal methods have a steep implementation curve and, to date, no easily accessible open-source software framework has been available for their systematic implementation. Methods: In this article, we introduce poly_rust, a new software library implementing a domain-specific language (DSL) specifically designed for polytopal methods. poly_rust provides high-level constructs for expressing a wide range of such methods using a syntax that closely mirrors their mathematical formulation, thereby substantially lowering the implementation barrier for both newcomers and experienced users. Results: The library facilitates the rapid implementation, testing, and comparison of polytopal discretizations, making these methods more accessible to researchers, applied scientists, engineers, and students. In our vision, this will foster their dissemination beyond the numerical analysis community and encourage their adoption by practitioners in a broad range of application fields. It will also contribute to reproducibility of results, as DSL files are text files that can be easily made available, stored, and exchanged. Conclusions: poly_rust introduces, to our knowledge, the first DSL specifically tailored to polytopal methods. By combining mathematical expressiveness, ease of implementation, and an open-source framework, it provides a common platform for developing, sharing, reproducing, and comparing polytopal schemes, with the ambition of becoming a reference tool for the wider scientific community.

math.NA↗

arXiv2026

A Reynolds- and Hartmann-semirobust hybrid method for magnetohydrodynamics

We propose and analyze a new method for the unsteady incompressible magnetohydrodynamics equations on convex domains with hybrid approximations of both vector-valued and scalar-valued fields. The proposed method is convection-semirobust, meaning that, for sufficiently smooth solutions, one can derive a priori estimates for the velocity and the magnetic field that do not depend on the inverse of the diffusion coefficients. This is achieved while at the same time providing relevant additional features, namely an improved order of convergence for the (asymptotic) diffusion-dominated regime, a small stencil (owing to the absence of inter-element penalty terms), and the possibility to significantly reduce the size of the algebraic problems through static condensation. The theoretical results are confirmed by a complete panel of numerical experiments.

math.NA↗

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.

math.NA↗

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.

math.NA↗

arXiv2026

A Reynolds-semi-robust H(div)-conforming method for unsteady incompressible power-law flows

In this work, we prove what appear to be the first Reynolds-semi-robust and pressure-robust velocity error estimates for an H(div)-conforming approximation of unsteady incompressible flows of power-law type fluids. The proposed methods hinges on a discontinuous Galerkin approximation of the viscous term and a reinforced upwind-type stabilization of the convective term. The derived velocity error estimates account for pre-asymptotic orders of convergence observed in convection-dominated flows through regime-dependent estimates of the error contributions. A complete set of numerical results validate the theoretical findings.

math.NA↗

arXiv2026

Optimal $hp$-error estimates and $p$-multigrid convergence for Hybrid High-Order discretizations of the Poisson equation

This paper presents two new theoretical results for Hybrid High-Order (HHO) methods applied to elliptic problems. First, we establish $hp$-error estimates for the HHO discretization of the Poisson problem that achieve optimal approximation rates with respect to both the mesh size $h$ and the polynomial degree $k$. These results improve upon previous analyses of hybrid methods, whose convergence estimates were suboptimal in $k$. Second, building on these estimates, we develop and analyze a non-inherited $p$-multigrid solver for the statically condensed HHO system. We prove results that improve upon the corresponding theory available for other non-conforming methods and constitute, to the best of our knowledge, the first rigorous convergence analysis of a $p$-multigrid algorithm for HHO discretizations.

math.NA↗

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.

math.NA↗

arXiv2026

A pressure-robust HHO method for the solution of the incompressible Navier-Stokes equations on general meshes

In a recent work [10], we have introduced a pressure-robust Hybrid High-Order method for the numerical solution of the incompressible Navier-Stokes equations on matching simplicial meshes. Pressure-robust methods are characterized by error estimates for the velocity that are fully independent of the pressure. A crucial question was left open in that work, namely whether the proposed construction could be extended to general polytopal meshes. In this paper we provide a positive answer to this question. Specifically, we introduce a novel divergence-preserving velocity reconstruction that hinges on the solution inside each element of a mixed problem on a subtriangulation, then use it to design discretizations of the body force and convective terms that lead to pressure robustness. An in-depth theoretical study of the properties of this velocity reconstruction, and their reverberation on the scheme, is carried out for polynomial degrees $k \geq 0$ and meshes composed of general polytopes. The theoretical convergence estimates and the pressure robustness of the method are confirmed by an extensive panel of numerical examples.

math.NA↗

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.

math.NA↗

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.

math.NA↗

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).

math.NA↗

arXiv2025

A Hybrid-High Order method for fracture modelling

In this work, we introduce a new Hybrid High-Order method for the numerical simulation of fracture propagation based on phase-field models. The proposed method supports general meshes made of polygonal/polyhedral elements, which provides great flexibility in mesh design and adaptation, and can accommodate large variations of both the displacement and damage variables thanks to the use of fully discontinuous spaces. The resolution of the corresponding algebraic problem is based on a staggered time stepping scheme which takes advantage of static condensation for each subproblem. We provide extensive numerical validation of the method on classical two-dimensional fracture propagation problems, including a comparison with a more standard finite element scheme.

math.NA↗

arXiv2025

Hyperelastic nature of the Hoek-Brown criterion

We propose a nonlinear elasto-plastic model, for which a specific class of hyperbolic elasticity arises as a straight consequence of the yield criterion invariance on the plasticity level. We superimpose this nonlinear elastic (or hyperelastic) behavior with plasticity obeying the associated flow rule. Interestingly, we find that a linear yield criterion on the thermodynamical force associated with plasticity results in a quadratic yield criterion in the stress space. This suggests a specific hyperelastic connection between Mohr-Coulomb and Hoek-Brown (or alternatively between Drucker-Prager and Pan-Hudson) yield criteria. We compare the elasto-plastic responses of standard tests for the Drucker-Prager yield criterion using either linear or the suggested hyperbolic elasticity. Notably, the nonlinear case stands out due to dilatancy saturation observed during cyclic loading in the triaxial compression test. We conclude this study with structural finite element simulations that clearly demonstrate the numerical applicability of the proposed model.

physics.app-ph↗

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.

math.NA↗

arXiv2025

An exterior calculus framework for polytopal methods

We develop in this work the first polytopal complexes of differential forms. These complexes, inspired by the Discrete De Rham and the Virtual Element approaches, are discrete versions of the de Rham complex of differential forms built on meshes made of general polytopal elements. Both constructions benefit from the high-level approach of polytopal methods, which leads, on certain meshes, to leaner constructions than the finite element method. We establish commutation properties between the interpolators and the discrete and continuous exterior derivatives, prove key polynomial consistency results for the complexes, and show that their cohomologies are isomorphic to the cohomology of the continuous de Rham complex.

math.NA↗

arXiv2024

A discrete de Rham discretization of interface diffusion problems with application to the Leaky Dielectric Model

Motivated by the study of the electrodynamics of particles, we propose in this work an arbitrary-order discrete de Rham scheme for the treatment of elliptic problems with potential and flux jumps across a fixed interface. The scheme seamlessly supports general elements resulting from the cutting of a background mesh along the interface. Interface conditions are enforced weakly à la Nitsche. We provide a rigorous convergence of analysis of the scheme for a steady model problem and showcase an application to a physical problem inspired by the Leaky Dielectric Model.

math.NA↗

arXiv2024

A Reynolds-semi-robust and pressure robust Hybrid High-Order method for the time dependent incompressible Navier--Stokes equations on general meshes

In this work we develop and analyze a Reynolds-semi-robust and pressure-robust Hybrid High-Order (HHO) discretization of the incompressible Navier--Stokes equations. Reynolds-semi-robustness refers to the fact that, under suitable regularity assumptions, the right-hand side of the velocity error estimate does not depend on the inverse of the viscosity. This property is obtained here through a penalty term which involves a subtle projection of the convective term on a subgrid space constructed element by element. The estimated convergence order for the $L^\infty(L^2)$- and $L^2(\text{energy})$-norm of the velocity is $h^{k+\frac12}$, which matches the best results for continuous and discontinuous Galerkin methods and corresponds to the one expected for HHO methods in convection-dominated regimes. Two-dimensional numerical results on a variety of polygonal meshes complete the exposition.

math.NA↗