arXiv subjects

Umberto Zerbinati

Publications and source records attributed to Umberto Zerbinati.

12 recordsLinked to original sources

arXiv2026

Preconditioned normal equations for solving discretised partial differential equations

This paper explores preconditioning the normal equation for non-symmetric square linear systems arising from PDE discretization, focusing on methods like CGNE and LSQR. The concept of ``normal'' preconditioning is introduced and a strategy to construct preconditioners studying the associated ``normal'' PDE is presented. Numerical experiments on convection-diffusion problems demonstrate the effectiveness of this approach in achieving fast and stable convergence.

math.NA↗

arXiv2026

Analysis and numerical analysis of the Helmholtz-Korteweg equation

We analyse the nematic Helmholtz-Korteweg equation, a variant of the classical Helmholtz equation that describes time-harmonic wave propagation in calamitic fluids in the presence of nematic order. A prominent example is given by nematic liquid crystals, which can be modeled as nematic Korteweg fluids - that is, fluids whose stress tensor depends on density gradients and on a nematic director describing the orientation of the anisotropic molecules. These materials exhibit anisotropic acoustic properties that can be tuned by external electromagnetic fields, making them attractive for potential applications such as tunable acoustic resonators. We prove the existence and uniqueness of solutions to this equation in two and three dimensions for suitable (nonresonant) wave numbers and propose a convergent discretisation for its numerical solution. The discretisation of this problem is nontrivial as it demands high regularity and involves unfamiliar boundary conditions. We address these challenges by using high-order conforming finite elements and enforcing the boundary conditions with Nitsche's method. We illustrate our analysis with numerical simulations in two dimensions.

math.NA↗

arXiv2026

Achieving Material Robustness via Symmetric Stress Finite Element Discretizations

When discretizing symmetric stress tensors in variational problems arising in continuum mechanics, one has to choose how to enforce the symmetry of the stress tensor: (i) strongly by requiring the discrete tensors to be pointwise symmetric or (ii) weakly by introducing a Lagrange multiplier. For $H(\mathrm{div})$-conforming finite element discretizations of Hellinger--Reissner elasticity and velocity--stress formulations of incompressible flow, where symmetry of the Cauchy stress tensor is tied to the conservation of angular momentum, we show that this choice may substantially impact the accuracy of the numerical scheme. Through a series of benchmark problems featuring anisotropic constitutive laws inspired by fiber reinforced material, liquid crystal polymer networks, and polar fluids, we show that schemes enforcing symmetry weakly can yield arbitrarily poor stress approximations -- even for zero-stress configurations. However, schemes enforcing symmetry strongly deliver accurate stress approximations independently of the constitutive law, a property we term material robustness. We present a unifying theory that rigorously explains this behavior.

math.NA↗

arXiv2026

A kinetic interpretation of thermomechanical restrictions of continua

Rajagopal and Srinivasa's thermodynamic framework derives constitutive relations in continuum mechanics from two scalar functions describing energy storage and entropy production via a constrained optimization principle. In parallel, kinetic theory obtains constitutive laws through moment closure, most notably via the Chapman--Enskog expansion. This work has three objectives. First, we establish a connection between these approaches by providing a kinetic interpretation of the Rajagopal--Srinivasa principle of maximal entropy production, under appropriate albeit restrictive hypotheses. For a Bhatnagar--Gross--Krook-type approximation, we show that the Rajagopal--Srinivasa principle is equivalent to a minimal relaxation-time principle, selecting among admissible constitutive responses the one with the fastest compatible relaxation toward equilibrium. Second, we review the classical kinetic description of continua in a manner accessible to those familiar with continuum thermodynamics. Third, we propose a hybrid Chapman--Enskog--Rajagopal--Srinivasa approach which computes the thermodynamic relations and entropy production from the Chapman--Enskog expansion, and then invokes the Rajagopal--Srinivasa principle to determine the other constitutive relations. This recovers the standard Euler and Navier--Stokes--Fourier constitutive laws for monatomic gases. We also demonstrate how different choices of selection procedure can be more informative than the classical Chapman--Enskog closure in the context of an inviscid compressible Leslie--Ericksen model arising in liquid crystals.

math-ph↗

arXiv2026

A Kinetic Theory Approach to Ordered Fluids

We develop a unified kinetic theory for ordered fluids, which systematically extends the phase space with the appropriate generalized angular momenta. Our theory yields a uniquely determined mesoscopic model for any continuum with microstructure that is characterized by Capriz's order parameter manifold. We illustrate our theory with three running examples: liquids saturated with non-diffusive gas bubbles, liquids composed of calamitic (rodlike) molecules, and liquids composed of calamitic molecules with additional head-to-tail symmetry. We discuss the symmetries of the microscopic interactions via Noether's theorem, and use them to characterize the conserved quantities mesoscopic dynamics. We derive the mesoscopic model for ordered fluids from a kinetic point of view assuming that the microscopic interactions are of weak nature when it comes to the ordering of the fluid. Lastly, we discuss under which conditions an H-theorem result holds at the mesoscopic scale and for which Vlasov potential we can expect the emergence of collective behavior.

math-ph↗

arXiv2025

Time-harmonic waves in Korteweg and nematic-Korteweg fluids

We derive the Helmholtz--Korteweg equation, which models acoustic waves in Korteweg fluids. We further derive a nematic variant of the Helmholtz-Korteweg equation, which incorporates an additional orientational term in the stress tensor. Its dispersion relation coincides with that arising in Virga's analysis of the Euler-Korteweg equations, which we extend to consider imaginary wave numbers and the effect of boundary conditions. In particular, our extensions allow us to analyze the effect of nematic orientation on the penetration depth of evanescent plane waves, and on the scattering of sound waves by obstacles. Furthermore, we make new, experimentally-verifiable predictions for the effect of boundary conditions for a modification of the Mullen-Lüthi-Stephen experiment, and for the scattering of acoustic waves in nematic-Korteweg fluids by a circular obstacle.

math-ph↗

arXiv2024

An adaptive mesh refinement strategy to ensure quasi-optimality of finite element methods for self-adjoint Helmholtz problems

It is well known that the quasi-optimality of the Galerkin finite element method for the Helmholtz equation is dependent on the mesh size and the wave-number. In the literature, different criteria have been proposed to ensure uniform quasi-optimality of the discretisation. In the present work, we study the uniform quasi-optimality of $H^1$ conforming and non-conforming Crouzeix-Raviart discretisation of the self-adjoint Helmholtz problem. In particular, we propose an adaptive scheme, coupled with a residual-based indicator, for generating guaranteed quasi-optimal meshes with minimal degrees of freedom.

math.NA↗

arXiv2024

A nodal ghost method based on variational formulation and regular square grid for elliptic problems on arbitrary domains in two space dimensions

This paper focuses on the numerical solution of elliptic partial differential equations (PDEs) with Dirichlet and mixed boundary conditions, specifically addressing the challenges arising from irregular domains. Both finite element method (FEM) and finite difference method (FDM), face difficulties in dealing with arbitrary domains. The paper introduces a novel nodal symmetric ghost {method based on a variational formulation}, which combines the advantages of FEM and FDM. The method employs bilinear finite elements on a structured mesh and provides a detailed implementation description. A rigorous a priori convergence rate analysis is also presented. The convergence rates are validated with many numerical experiments, in both one and two space dimensions.

math.NA↗

arXiv2024

Kinetic derivation of a compressible Leslie--Ericksen equation for rarified calamitic gases

Nematic ordering describes the phenomenon where anisotropic molecules tend to locally align, like matches in a matchbox. This ordering can arise in solids (as nematic elastomers), liquids (as liquid crystals), and in gases. In the 1940s, Onsager described how nematic ordering can arise in dilute colloidal suspensions from the molecular point of view. However, the kinetic theory of nonspherical molecules has not, thus far, accounted for phenomena relating to the presence of nematic ordering. In this work we develop a kinetic theory for the behavior of rarified calamitic (rodlike) gases in the presence of nematic ordering. Building on previous work by Curtiss, we derive from kinetic theory the rate of work hypothesis that forms the starting point for Leslie--Ericksen theory. We incorporate ideas from the variational theory of nematic liquid crystals to create a moment closure that preserves the coupling between the laws of linear and angular momentum. The coupling between these laws is a key feature of our theory, in contrast to the kinetic theory proposed by {Curtiss \& Dahler}, where the couple stress tensor is assumed to be zero. This coupling allows the characterization of anisotropic phenomena arising from the nematic ordering. Furthermore, the theory leads to an energy functional that is a compressible variant of the classical Oseen--Frank energy (with a pressure-dependent Frank constant) and to a compressible analogue of the Leslie--Ericksen equations. The emergence of compressible aspects in the theory for nematic fluids enhances our understanding of these complex systems.

math-ph↗

arXiv2024

A comparison of the Coco-Russo scheme and $\protect\mathghost$-FEM for elliptic equations in arbitrary domains

In this paper, a comparative study between the Coco-Russo scheme (based on finite-difference scheme) and the $\mathghost$-FEM (based on finite-element method) is presented when solving the Poisson equation in arbitrary domains. The comparison between the two numerical methods is carried out by presenting analytical results from the literature \cite{cocoStissi,astuto2024nodal}, together with numerical tests in various geometries and boundary conditions.

math.NA↗

arXiv2024

PETScML: Second-order solvers for training regression problems in Scientific Machine Learning

In recent years, we have witnessed the emergence of scientific machine learning as a data-driven tool for the analysis, by means of deep-learning techniques, of data produced by computational science and engineering applications. At the core of these methods is the supervised training algorithm to learn the neural network realization, a highly non-convex optimization problem that is usually solved using stochastic gradient methods. However, distinct from deep-learning practice, scientific machine-learning training problems feature a much larger volume of smooth data and better characterizations of the empirical risk functions, which make them suited for conventional solvers for unconstrained optimization. We introduce a lightweight software framework built on top of the Portable and Extensible Toolkit for Scientific computation to bridge the gap between deep-learning software and conventional solvers for unconstrained minimization. We empirically demonstrate the superior efficacy of a trust region method based on the Gauss-Newton approximation of the Hessian in improving the generalization errors arising from regression tasks when learning surrogate models for a wide range of scientific machine-learning techniques and test cases. All the conventional second-order solvers tested, including L-BFGS and inexact Newton with line-search, compare favorably, either in terms of cost or accuracy, with the adaptive first-order methods used to validate the surrogate models.

cs.LG↗

arXiv2022

PINNs and GaLS: A Priori Error Estimates for Shallow Physics Informed Neural Networks Applied to Elliptic Problems

Physics Informed Neural Networks (PINNs) have recently gained popularity for solving partial differential equations, given the fact they escape the curse of dimensionality. In this paper, we present Physics Informed Neural Networks as an underdetermined point matching collocation method then expose the connection between Galerkin Least Square (GALS) and PINNs, to develop an a priori error estimate, in the context of elliptic problems. In particular, techniques that belong to the realm of least square finite elements and Rademacher complexity analysis are used to obtain the error estimate.

math.NA↗