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Giuseppe Vacca

Publications and source records attributed to Giuseppe Vacca.

At least 19 recordsLinked to original sources
arXiv2026

A Mixed Virtual Element Method for the p-Laplace equation

We introduce and analyze a mixed Virtual Element Method for the $p$-Laplace equation in a non-Hilbertian setting, covering the full range $p \in (1, \infty)$. The discrete framework combines standard mixed Virtual Element spaces with a novel non-linear stabilization term designed to mimic the power-law structure of the continuous operator. We establish discrete inf-sup stability under non-Hilbertian norms and rigorously prove the continuity and coercivity of the discrete form. This guarantees the well-posedness of the problem and allows us to derive a priori error estimates for the primal variable and the flux. A set of numerical tests supports the theoretical derivations.

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arXiv2026

Divergence-free and mass-conservative virtual element methods for the Navier-Stokes-Cahn-Hilliard system

In this work, we design and analyze semi/fully-discrete virtual element approximations for the time-dependent Navier--Stokes-Cahn--Hilliard equations, modeling the dynamics of two-phase incompressible fluid flows with diffuse interfaces. A new variational formulation is derived involving solely the velocity, pressure, and phase field, together with corresponding a priori energy estimates. The spatial discretization is based on the coupling divergence-free and $C^1$-conforming elements of high-order, while the time discretization employs a classical backward Euler scheme. By introducing a novel skew-symmetric trilinear form to discretize the convective term in the Cahn--Hilliard equation, we propose discrete schemes that satisfy mass conservation and energy bounds. Moreover, optimal error estimates are provided for both formulations. Finally, two numerical experiments are presented to support our theoretical findings and to illustrate the good performance of the proposed schemes for different polynomial degrees and polygonal meshes.

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arXiv2026

Virtual Element methods for non-Newtonian shear-thickening fluid flow problems

In this work, we present a comprehensive theoretical analysis for Virtual Element discretizations of incompressible non-Newtonian flows governed by the Carreau-Yasuda constitutive law, in the shear-thickening regime (r > 2) including both degenerate (delta = 0) and non-degenerate (delta > 0) cases. The proposed Virtual Element method features two distinguishing advantages: the construction of an exactly divergence-free discrete velocity field and compatibility with general polygonal meshes. The analysis presented in this work extends a previous work, where only shear-thinning behavior (1 < r < 2) was considered. Indeed, the theoretical analysis of the shear-thickening setting requires several novel analytical tools, including: an inf-sup stability analysis of the discrete velocity-pressure coupling in non-Hilbertian norms, a stabilization term specifically designed to address the nonlinear structure as the exponent r > 2; and the introduction of a suitable discrete norm tailored to the underlying nonlinear constitutive relation. Numerical results demonstrate the practical performance of the proposed formulation.

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arXiv2025

Stability and interpolation estimates of Hellinger-Reissner virtual element spaces

We prove stability and interpolation estimates for Hellinger-Reissner virtual elements; the constants appearing in such estimates only depend on the aspect ratio of the polytope under consideration and the degree of accuracy of the scheme. We further investigate numerically the behaviour of the constants appearing in the stability estimates on sequences of badly-shaped polytopes and for increasing degree of accuracy.

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arXiv2022

Mixed Virtual Element approximation of linear acoustic wave equation

We design a Mixed Virtual Element Method for the approximated solution to the first-order form of the acoustic wave equation. In absence of external load, the semi-discrete method exactly conserves the system energy. To integrate in time the semi-discrete problem we consider a classical theta-method scheme. We carry out the stability and convergence analysis in the energy norm for the semi-discrete problem showing optimal rate of convergence with respect to the mesh size. We further study the property of energy conservation for the fully-discrete system. Finally, we present some verification tests as well as engineering application of the method.

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arXiv2022

Virtual Element Method for the Navier--Stokes Equation coupled with the Heat Equation

We consider the Virtual Element discretization of the Navier-Stokes equations coupled with the heat equation where the viscosity depends on the temperature. We present the Virtual Element discretization of the coupled problem, show its well-posedness, and prove optimal error estimates. Numerical experiments which confirm the theoretical error bounds are also presented.

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arXiv2021

$C^1$-VEM for some variants of the Cahn-Hilliard equation: a numerical exploration

We consider the $C^1$-Virtual Element Method (VEM) for the conforming numerical approximation of some variants of the Cahn-Hilliard equation on polygonal meshes. In particular, we focus on the discretization of the advective Cahn-Hilliard problem and the Cahn-Hilliard inpainting problem. We present the numerical approximation and several numerical results to assess the efficacy of the proposed methodology.

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arXiv2021

Bend 3d Mixed Virtual Element Method for Elliptic Problems

In this study, we propose a virtual element scheme to solve the Darcy problem in three physical dimensions. The main novelty, here proposed, is that curved elements are naturally handled without any degradation of the solution accuracy. In fact, in presence of curved boundaries, or internal interfaces, the geometrical error introduced by planar approximations may dominate the convergence rate limiting the benefit of high-order approximations. We consider the Darcy problem in its mixed form to directly obtain, with our numerical scheme, accurate and mass conservative fluxes without any post-processing. An important step to derive this new scheme is the actual computation of polynomials over curved polyhedrons, here presented and discussed. Finally, we show the theoretical analysis of the scheme as well as several numerical examples to support our findings

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arXiv2021

A Virtual Element Method for the wave equation on curved edges in two dimensions

In this work we present an extension of the Virtual Element Method with curved edges for the numerical approximation of the second order wave equation in a bidimensional setting. Curved elements are used to describe the domain boundary, as well as internal interfaces corresponding to the change of some mechanical parameters. As opposite to the classic and isoparametric Finite Element approaches, where the geometry of the domain is approximated respectively by piecewise straight lines and by higher order polynomial maps, in the proposed method the geometry is exactly represented, thus ensuring a highly accurate numerical solution. Indeed, if in the former approach the geometrical error might deteriorate the quality of the numerical solution, in the latter approach the curved interfaces/boundaries are approximated exactly guaranteeing the expected order of convergence for the numerical scheme. Theoretical results and numerical findings confirm the validity of the proposed approach.

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arXiv2020

The mixed virtual element method for grids with curved interfaces

In many applications the accurate representation of the computational domain is a key factor to obtain reliable and effective numerical solutions. Curved interfaces, which might be internal, related to physical data, or portions of the physical boundary, are often met in real applications. However, they are often approximated leading to a geometrical error that might become dominant and deteriorate the quality of the results. Underground problems often involve the motion of fluids where the fundamental governing equation is the Darcy law. High quality velocity fields are of paramount importance for the successful subsequent coupling with other physical phenomena such as transport. The virtual element method, as solution scheme, is known to be applicable in problems whose discretizations requires cells of general shape, and the mixed formulation is here preferred to obtain accurate velocity fields. To overcome the issues associated to the complex geometries and, at the same time, retaining the quality of the solutions, we present here the virtual element method to solve the Darcy problem, in mixed form, in presence of curved interfaces in two and three dimensions. The numerical scheme is presented in detail explaining the discrete setting with a focus on the treatment of curved interfaces. Examples, inspired from industrial applications, are presented showing the validity of the proposed approach.

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arXiv2020

The Mixed Virtual Element Method on curved edges in two dimensions

In this work, we propose an extension of the mixed Virtual Element Method (VEM) for bi-dimensional computational grids with curvilinear edge elements. The approximation by means of rectilinear edges of a domain with curvilinear geometrical feature, such as a portion of domain boundary or an internal interface, may introduce a geometrical error that degrades the expected order of convergence of the scheme. In the present work a suitable VEM approximation space is proposed to consistently handle curvilinear geometrical objects, thus recovering optimal convergence rates. The resulting numerical scheme is presented along with its theoretical analysis and several numerical test cases to validate the proposed approach.

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arXiv2019

A virtual element method for the miscible displacement of incompressible fluids in porous media

In the present contribution, we construct a virtual element (VE) discretization for the problem of miscible displacement of one incompressible fluid by another, described by a time-dependent coupled system of nonlinear partial differential equations. Our work represents a first study to investigate the premises of virtual element methods (VEM) for complex fluid flow problems. We combine the VEM discretization with a time stepping scheme and develop a complete theoretical analysis of the method under the assumption of a regular solution. The scheme is then tested both on a regular and on a more realistic test case.

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arXiv2018

Bricks for the mixed high-order virtual element method: projectors and differential operators

We present the essential instruments to deal with Virtual Element Method (VEM) for the resolution of partial differential equations in mixed form. Functional spaces, degrees of freedom, projectors and differential operators are described emphasizing how to build them in a virtual element framework and for a general approximation order. To achieve this goal, it was necessary to make a deep analysis on polynomial spaces and decompositions. Finally, we exploit such `briks' to construct virtual element approximations of Stokes, Darcy and Navier-Stokes problems and we provide a series of examples to numerically verify the theoretical behavior of high-order VEM.

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arXiv2018

The virtual element method for eigenvalue problems with potential terms on polytopal meshes

We extend the conforming virtual element method to the numerical resolution of eigenvalue problems with potential terms on a polytopal mesh. An important application is that of the Schrodinger equation with a pseudopotential term. This model is a fundamental element in the numerical resolution of more complex problems from the Density Functional Theory. The VEM is based on the construction of the discrete bilinear forms of the variational formulation through certain polynomial projection operators that are directly computable from the degrees of freedom. The method shows a great flexibility with respect to the meshes and provide a correct spectral approximation with optimal convergence rates. This point is discussed from both the theoretical and the numerical viewpoint. The performance of the method is numerically investigated by solving the Quantum Harmonic Oscillator problem with the harmonic potential and a singular eigenvalue problem with zero potential for the first eigenvalues.

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arXiv2018

The nonconforming virtual element method for eigenvalue problems

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allow to treat in the same formulation the two- and three-dimensional case.We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of the L^2-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

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arXiv2017

An H^1-conforming Virtual Element Method for Darcy equations and Brinkman equations

The focus of the present paper is on developing a Virtual Element Method for Darcy and Brinkman equations. In [15] we presented a family of Virtual Elements for Stokes equations and we defined a new Virtual Element space of velocities such that the associated discrete kernel is pointwise divergence-free. We use a slightly different Virtual Element space having two fundamental properties: the L^2-projection onto P_k is exactly computable on the basis of the degrees of freedom, and the associated discrete kernel is still pointwise divergence-free. The resulting numerical scheme for the Darcy equation has optimal order of convergence and H^1 conforming velocity solution. We can apply the same approach to develop a robust virtual element method for the Brinkman equation that is stable for both the Stokes and Darcy limit case. We provide a rigorous error analysis of the method and several numerical tests.

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arXiv2016

Virtual Element Methods for hyperbolic problems on polygonal meshes

In the present paper we develop the Virtual Element Method for hyperbolic problems on polygonal meshes, considering the linear wave equations as our model problem. After presenting the semi-discrete scheme, we derive the convergence estimates in H^1 semi-norm and L^2 norm. Moreover we develop a theoretical analysis on the stability for the fully discrete problem by comparing the Newmark method and the Bathe method. Finally we show the practical behaviour of the proposed method through a large array of numerical tests.

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