arXiv subjects

F. Dassi

Publications and source records attributed to F. Dassi.

17 recordsLinked to original sources

arXiv2026

Conforming/Non-conforming Virtual Elements and application to elasticity problems in curved three-dimensional domains

The Virtual Element Method (VEM) is a well-established framework for solving partial differential equations on polygonal and polyhedral meshes. In this paper, we introduce a novel hybrid VEM that integrates both conforming and nonconforming virtual spaces. We apply this formulation to a three-dimensional linear elasticity problem, providing rigorous theoretical analysis to demonstrate optimal convergence rates. Furthermore, we explore the extension of this approach to domains with curved boundaries.

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arXiv2025

Fully and semi-implicit robust space-time DG methods for the incompressible Navier-Stokes equations

We carry out a stability and convergence analysis of a fully discrete scheme for the time-dependent Navier-Stokes equations resulting from combining an $H(\mathrm{div}, \Omega)$-conforming discontinuous Galerkin spatial discretization, and a discontinuous Galerkin time stepping scheme. Such a scheme is proven to be pressure robust and Reynolds semi-robust. Standard techniques can be used to analyze only the case of lowest-order approximations in time. Therefore, we use some nonstandard test functions to prove existence of discrete solutions, unconditional stability, and quasi-optimal convergence rates for any degree of approximation in time. In particular, a continuous dependence of the discrete solution on the data of the problem, and quasi-optimal convergence rates for low and high Reynolds numbers are proven in an energy norm including the term $L^{\infty}(0, T; L^2(\Omega)^d)$ for the velocity. In addition to the standard discontinuous Galerkin time stepping scheme, which is fully implicit, we propose and analyze a novel high-order semi-implicit version that avoids the need of solving nonlinear systems of equations after the first time slab, thus significantly improving the efficiency of the method. Some numerical experiments validating our theoretical results are presented.

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arXiv2024

Pressure and convection robust Finite Elements for Magnetohydrodynamics

We propose and analyze two convection quasi-robust and pressure robust finite element methods for a fully nonlinear time-dependent magnetohydrodynamics problem. Both methods employ the $H_{\rm div}$ conforming BDM element coupled with an appropriate pressure space guaranteeing the exact diagram for the fluid part, and the $H^1$ conforming Lagrange element for the approximation of the magnetic fluxes, and make use of suitable DG upwind terms and CIP stabilizations to handle the fluid and magnetic convective terms. The main difference between the two approaches here proposed (labeled as three-field scheme and four field-scheme respectively) lies in the strategy adopted to enforce the divergence-free condition of the magnetic field. The three-filed scheme implements a grad-div stabilization, whereas the four-field scheme introduces a suitable Lagrange multiplier and additional stabilization terms in the formulation. The developed error estimates for the two schemes are uniform in both diffusion parameters and optimal with respect to the diffusive norm. Furthermore, in the convection dominated regime, being $k$ the degree of the method and $h$ the mesh size, we are able to prove $O(h^k)$ and $O(h^{k+1/2})$ pre-asymptotic error reduction rate for the three-field scheme and four-filed scheme respectively. A set of numerical tests support our theoretical findings.

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arXiv2023

Vem++, a c++ library to handle and play with the Virtual Element Method

The Virtual Element Method (VEM) is an extension of the Finite Element Method (FEM) used for handling polytopal meshes. This paper provides a brief introduction to the VEM for a two-dimensional Laplacian problem. Additionally, it highlights the differences between a VEM implementation and a FEM code, emphasising the main challenges associated with the VEM. Furthermore, this paper presents a possible approach to address these challenges: vem++, a c++ library specifically developed for working with VEM discretisation. The library is designed to handle various partial differential equations in two or three dimensions, arising from both academic and engineering problems. Its flexible design allows for the seamless integration of new features, such as novel polytopes' quadrature rules, solvers, and virtual element spaces.

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arXiv2023

Robust Finite Elements for linearized Magnetohydrodynamics

We introduce a pressure robust Finite Element Method for the linearized Magnetohydrodynamics equations in three space dimensions, which is provably quasi-robust also in the presence of high fluid and magnetic Reynolds numbers. The proposed scheme uses a non-conforming BDM approach with suitable DG terms for the fluid part, combined with an $H^1$-conforming choice for the magnetic fluxes. The method introduces also a specific CIP-type stabilization associated to the coupling terms. Finally, the theoretical result are further validated by numerical experiments.

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arXiv2023

Pressure robust SUPG-stabilized finite elements for the unsteady Navier-Stokes equation

In the present contribution we propose a novel conforming Finite Element scheme for the time-dependent Navier-Stokes equation, which is proven to be both convection quasi-robust and pressure robust. The method is built combining a "divergence-free" velocity/pressure couple (such as the Scott-Vogelius element), a Discontinuous Galerkin in time approximation, and a suitable SUPG-curl stabilization. A set of numerical tests, in accordance with the theoretical results, is included.

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arXiv2022

Machine Learning based refinement strategies for polyhedral grids with applications to Virtual Element and polyhedral Discontinuous Galerkin methods

We propose two new strategies based on Machine Learning techniques to handle polyhedral grid refinement, to be possibly employed within an adaptive framework. The first one employs the k-means clustering algorithm to partition the points of the polyhedron to be refined. This strategy is a variation of the well known Centroidal Voronoi Tessellation. The second one employs Convolutional Neural Networks to classify the "shape" of an element so that "ad-hoc" refinement criteria can be defined. This strategy can be used to enhance existing refinement strategies, including the k-means strategy, at a low online computational cost. We test the proposed algorithms considering two families of finite element methods that support arbitrarily shaped polyhedral elements, namely the Virtual Element Method (VEM) and the Polygonal Discontinuous Galerkin (PolyDG) method. We demonstrate that these strategies do preserve the structure and the quality of the underlaying grids, reducing the overall computational cost and mesh complexity.

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arXiv2021

Adaptive virtual elements based on hybridized, reliable, and efficient flux reconstructions

We present two a posteriori error estimators for the virtual element method (VEM) based on global and local flux reconstruction in the spirit of [5]. The proposed error estimators are reliable and efficient for the $h$-, $p$-, and $hp$-versions of the VEM. This solves a partial limitation of our former approach in [6], which was based on solving local nonhybridized mixed problems. Differently from the finite element setting, the proof of the efficiency turns out to be simpler, as the flux reconstruction in the VEM does not require the existence of polynomial, stable, divergence right-inverse operators. Rather, we only need to construct right-inverse operators in virtual element spaces, exploiting only the implicit definition of virtual element functions. The theoretical results are validated by some numerical experiments on a benchmark problem.

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arXiv2021

Virtual elements for Maxwell's equations

We present a low order virtual element discretization for time dependent Maxwell's equations, which allow for the use of general polyhedral meshes. Both the semi- and fully-discrete schemes are considered. We derive optimal a priori estimates and validate them on a set of numerical experiments. As pivot results, we discuss some novel inequalities for de Rahm sequences of nodal, edge, and face virtual element spaces.

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arXiv2020

SUPG-stabilized Virtual Elements for diffusion-convection problems: a robustness analysis

The objective of this contribution is to develop a convergence analysis for SUPG-stabilized Virtual Element Methods in diffusion-convection problems that is robust also in the convection dominated regime. For the original method introduced in [Benedetto et al, CMAME 2016] we are able to show an "almost uniform" error bound (in the sense that the unique term that depends in an unfavorable way on the parameters is damped by a higher order mesh-size multiplicative factor). We also introduce a novel discretization of the convection term that allows us to develop error estimates that are fully robust in the convection dominated cases. We finally present some numerical result.

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arXiv2019

Curvilinear Virtual Elements for 2D solid mechanics applications

In the present work we generalize the curvilinear Virtual Element technology, introduced for a simple linear scalar problem in a previous work, to generic 2D solid mechanic problems in small deformations. Such generalization also includes the development of a novel Virtual Element space for displacements that contains rigid body motions. Our approach can accept a generic black-box (elastic or inelastic) constitutive algorithm and, in addition, can make use of curved edges thus leading to an exact approximation of the geometry. Rigorous theoretical interpolation properties for the new space on curvilinear elements are derived. We undergo an extensive numerical test campaign, both on elastic and inelastic problems, to assess the behavior of the scheme. The results are very promising and underline the advantages of the curved VEM approach over the standard one in the presence of curved geometries.

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arXiv2019

A three-dimensional Hellinger-Reissner Virtual Element Method for linear elasticity problems

We present a Virtual Element Method for the 3D linear elasticity problems, based on Hellinger-Reissner variational principle. In the framework of the small strain theory, we propose a low-order scheme with a-priori symmetric stresses and continuous tractions across element interfaces. A convergence and stability analysis is developed and we confirm the theoretical predictions via some numerical tests.

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arXiv2019

The Stokes complex for Virtual Elements in three dimensions

The present paper has two objectives. On one side, we develop and test numerically divergence free Virtual Elements in three dimensions, for variable ``polynomial'' order. These are the natural extension of the two-dimensional divergence free VEM elements, with some modification that allows for a better computational efficiency. We test the element's performance both for the Stokes and (diffusion dominated) Navier-Stokes equation. The second, and perhaps main, motivation is to show that our scheme, also in three dimensions, enjoys an underlying discrete Stokes complex structure. We build a pair of virtual discrete spaces based on general polytopal partitions, the first one being scalar and the second one being vector valued, such that when coupled with our velocity and pressure spaces, yield a discrete Stokes complex.

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arXiv2019

Parallel solvers for virtual element discretizations of elliptic equations in mixed form

The aim of this paper is twofold. On the one hand, we test numerically the performance of mixed virtual elements in three dimensions for the first time in the literature to solve the mixed formulation of three-dimensional elliptic equations on polyhedral meshes. On the other hand, we focus on the parallel solution of the linear system arising from such discretization, considering both direct and iterative parallel solvers. In the latter case, we develop two block preconditioners, one based on the approximate Schur complement and one on a regularization technique. Both these topics are numerically validated by several parallel tests performed on a Linux cluster. More specifically, we show that the proposed VEM discretization recovers the expected theoretical convergence properties and we analize the performance of the direct and iterative parallel solvers taken into account.

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arXiv2018

A family of three-dimensional virtual elements with applications to magnetostatic

We consider, as a simple model problem, the application of Virtual Element Methods (VEM) to the linear Magnetostatic three-dimensional problem in the formulation of F. Kikuchi. In doing so, we also introduce new serendipity VEM spaces, where the serendipity reduction is made only on the faces of a general polyhedral decomposition (assuming that internal degrees of freedom could be more easily eliminated by static condensation). These new spaces are meant, more generally, for the combined approximation of $H^1$-conforming ($0$-forms), $H({\rm {\bf curl}})$-conforming ($1$-forms), and $H({\rm div})$-conforming ($2$-forms) functional spaces in three dimensions, and they would surely be useful for other problems and in more general contexts.

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arXiv2017

Lowest order Virtual Element approximation of magnetostatic problems

We give here a simplified presentation of the lowest order Serendipity Virtual Element method, and show its use for the numerical solution of linear magneto-static problems in three dimensions. The method can be applied to very general decompositions of the computational domain (as is natural for Virtual Element Methods) and uses as unknowns the (constant) tangential component of the magnetic field $\mathbf{H}$ on each edge, and the vertex values of the Lagrange multiplier $p$ (used to enforce the solenoidality of the magnetic induction $\mathbf{B}=\mu\mathbf{H}$). In this respect the method can be seen as the natural generalization of the lowest order Edge Finite Element Method (the so-called "first kind N\'ed\'elec" elements) to polyhedra of almost arbitrary shape, and as we show on some numerical examples it exhibits very good accuracy (for being a lowest order element) and excellent robustness with respect to distortions.

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arXiv2017

High-order Virtual Element Method on polyhedral meshes

We develop a numerical assessment of the Virtual Element Method for the discretization of a diffusion-reaction model problem, for higher "polynomial" order k and three space dimensions. Although the main focus of the present study is to illustrate some h-convergence tests for different orders k, we also hint on other interesting aspects such as structured polyhedral Voronoi meshing, robustness in the presence of irregular grids, sensibility to the stabilization parameter and convergence with respect to the order k.

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