17 results on '"Giuseppe Vacca"'
Search Results
2. The p- and hp-versions of the virtual element method for elliptic eigenvalue problems
- Author
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Francesca Gardini, Giuseppe Vacca, Lorenzo Mascotto, Gianmarco Manzini, and Ondřej Čertík
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Degrees of freedom (statistics) ,Eigenfunction ,Mass matrix ,Schrödinger equation ,Exponential function ,Sobolev space ,Computational Mathematics ,Elliptic operator ,symbols.namesake ,Computational Theory and Mathematics ,Modeling and Simulation ,symbols ,Applied mathematics ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We discuss the p - and h p -versions of the virtual element method for the approximation of eigenpairs of elliptic operators with a potential term on polygonal meshes. An application of this model is provided by the Schrodinger equation with a pseudo-potential term. As an interesting byproduct, we present for the first time in literature an explicit construction of the stabilization of the mass matrix. We present in detail the analysis of the p -version of the method, proving exponential convergence in the case of analytic eigenfunctions. The theoretical results are supplied with a wide set of experiments. We also show numerically that, in the case of eigenfunctions with finite Sobolev regularity, an exponential approximation of the eigenvalues in terms of the cubic root of the number of degrees of freedom can be obtained by employing h p -refinements. Importantly, the geometric flexibility of polygonal meshes is exploited in the construction of the h p -spaces.
- Published
- 2020
3. A Virtual Element Method for the Wave Equation on Curved Edges in Two Dimensions
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Franco Dassi, Alessio Fumagalli, Ilario Mazzieri, Anna Scotti, Giuseppe Vacca, Dassi, F, Fumagalli, A, Mazzieri, I, Scotti, A, and Vacca, G
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Numerical Analysis ,Applied Mathematics ,General Engineering ,Numerical Analysis (math.NA) ,Theoretical Computer Science ,Computational Mathematics ,Computational Theory and Mathematics ,Polygonal grids ,FOS: Mathematics ,Curved elements ,Wave equation ,Curved element ,Mathematics - Numerical Analysis ,Polygonal grid ,Virtual element method ,Software - Abstract
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.
- Published
- 2022
4. Bend 3d mixed virtual element method for Darcy problems
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Franco Dassi, Alessio Fumagalli, Anna Scotti, Giuseppe Vacca, Dassi, F, Fumagalli, A, Scotti, A, and Vacca, G
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Computational Mathematics ,Integration over curved polyhedron ,Computational Theory and Mathematics ,Modeling and Simulation ,Mixed VEM ,High order approximation ,Curved face - Abstract
In this study, we propose a virtual element scheme to solve the Darcy problem in three physical dimensions. The main novelty is that curved elements are naturally handled without any degradation of the solution accuracy. Indeed, 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 accurate and mass conservative fluxes without any post-processing. An important step to derive the proposed scheme is the integration 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.
- Published
- 2022
5. SUPG-stabilized virtual elements for diffusion-convection problems: A robustness analysis
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Carlo Lovadina, Giuseppe Vacca, Franco Dassi, L. Beirão da Veiga, Beirao da Veiga, L, Dassi, F, Lovadina, C, and Vacca, G
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Convection ,Discretization ,010103 numerical & computational mathematics ,01 natural sciences ,SUPG stabilization ,Robustness (computer science) ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,Virtual element method ,0101 mathematics ,Mathematics ,Numerical Analysis ,Applied Mathematics ,Multiplicative function ,Diffusion convection ,Numerical Analysis (math.NA) ,Term (time) ,010101 applied mathematics ,Computational Mathematics ,Modeling and Simulation ,Convection dominated problem ,Element (category theory) ,Analysis ,Polygonal meshe - Abstract
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 unfavourable 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.
- Published
- 2021
6. Mimetic finite difference methods for Hamiltonian wave equations in 2D
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Giuseppe Vacca, Luciano Lopez, L. Beirão da Veiga, Beirao da Veiga, L, Lopez, L, and Vacca, G
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Discretization ,010103 numerical & computational mathematics ,01 natural sciences ,Mimetic finite difference method ,Hamiltonian system ,symbols.namesake ,Computational Theory and Mathematic ,FOS: Mathematics ,Covariant Hamiltonian field theory ,Mathematics - Numerical Analysis ,Hamiltonian systems ,0101 mathematics ,Mathematics::Symplectic Geometry ,Mathematics ,Mimetic finite difference methods ,Polygonal meshes ,Mathematical analysis ,Finite difference method ,Finite difference ,Numerical Analysis (math.NA) ,Wave equation ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Modeling and Simulation ,symbols ,Hamiltonian (quantum mechanics) ,Polygonal meshe ,Symplectic geometry - Abstract
In this paper we consider the numerical solution of the Hamiltonian wave equation in two spatial dimension. We use the Mimetic Finite Difference (MFD) method to approximate the continuous problem combined with a symplectic integration in time to integrate the semi-discrete Hamiltonian system. The main characteristic of MFD methods, when applied to stationary problems, is to mimic important properties of the continuous system. This approach, associated with a symplectic method for the time integration yields a full numerical procedure suitable to integrate Hamiltonian problems. A complete theoretical analysis of the method and some numerical simulations are developed in the paper., Comment: 26 pages, 8 figures
- Published
- 2017
7. Structural dynamical systems: Computational aspects
- Author
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Luca Gerardo-Giorda, Roberto Garrappa, Cinzia Elia, Alessandro Pugliese, and Giuseppe Vacca
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Computational Mathematics ,Numerical Analysis ,Dynamical systems theory ,Applied Mathematics ,Statistical physics ,Mathematics - Published
- 2020
8. The Stokes Complex for Virtual Elements with Application to Navier–Stokes Flows
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David Mora, L. Beirão da Veiga, Giuseppe Vacca, Beirao da Veiga, L, Mora, D, and Vacca, G
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Numerical Analysis ,Pure mathematics ,Discretization ,Applied Mathematics ,Computability ,General Engineering ,Degrees of freedom (physics and chemistry) ,Structure (category theory) ,010103 numerical & computational mathematics ,Differential operator ,01 natural sciences ,Discrete velocity ,Theoretical Computer Science ,Virtual elements ,010101 applied mathematics ,Computational Mathematics ,Computational Theory and Mathematics ,Navier stokes ,0101 mathematics ,Element (category theory) ,Software ,Mathematics ,Polygonal meshe ,Stokes complex - Abstract
In the present paper, we investigate the underlying Stokes complex structure of the Virtual Element Method for Stokes and Navier–Stokes introduced in previous papers by the same authors, restricting our attention to the two dimensional case. We introduce a Virtual Element space $${\varPhi }_h \subset H^2({\varOmega })$$ Φ h ⊂ H 2 ( Ω ) and prove that the triad $$\{{\varPhi }_h, {\varvec{V}}_h, Q_h\}$$ { Φ h , V h , Q h } (with $${\varvec{V}}_h$$ V h and $$Q_h$$ Q h denoting the discrete velocity and pressure spaces) is an exact Stokes complex. Furthermore, we show the computability of the associated differential operators in terms of the adopted degrees of freedom and explore also a different discretization of the convective trilinear form. The theoretical findings are supported by numerical tests.
- Published
- 2019
9. Bricks for the mixed high-order virtual element method: projectors and differential operators
- Author
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Franco Dassi, Giuseppe Vacca, Dassi, F, and Vacca, G
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Numerical Analysis ,Polynomial ,Partial differential equation ,Series (mathematics) ,Applied Mathematics ,Numerical analysis ,Degrees of freedom (statistics) ,Computational mathematics ,010103 numerical & computational mathematics ,Differential operator ,01 natural sciences ,Mixed problem ,010101 applied mathematics ,Algebra ,Computational Mathematics ,Computational Mathematic ,Virtual element method ,Mathematics - Numerical Analysis ,0101 mathematics ,Element (category theory) ,Projector ,High-order ,Numerical Analysi ,Polygonal meshe ,Mathematics - Abstract
We present the essential tools to deal with virtual element method (VEM) for the approximation of solutions 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. We exploit such “bricks” to construct virtual element approximations of Stokes, Darcy and Navier–Stokes problems and we provide a series of examples to numerically verify the theoretical behaviour of high-order VEM.
- Published
- 2018
10. The nonconforming virtual element method for eigenvalue problems
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Gianmarco Manzini, Giuseppe Vacca, Francesca Gardini, Gardini, F, Manzini, G, and Vacca, G
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Numerical Analysis ,Applied Mathematics ,Spectrum (functional analysis) ,010103 numerical & computational mathematics ,Numerical Analysis (math.NA) ,Eigenfunction ,01 natural sciences ,010101 applied mathematics ,Computational Mathematics ,Rate of convergence ,nonconforming virtual element, eigenvalue problem, polygonal meshes ,Large set (Ramsey theory) ,Modeling and Simulation ,Product (mathematics) ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Element (category theory) ,Analysis ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allows 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 theL2-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.
- Published
- 2018
- Full Text
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11. Virtual Elements for the Navier--Stokes Problem on Polygonal Meshes
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Carlo Lovadina, Giuseppe Vacca, L. Beirão da Veiga, Beirao da Veiga, L, Lovadina, C, and Vacca, G
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Field (physics) ,Mathematics::Analysis of PDEs ,010103 numerical & computational mathematics ,01 natural sciences ,Navier-Stokes equation ,Physics::Fluid Dynamics ,FOS: Mathematics ,Polygon mesh ,Navier stokes ,Virtual element method ,Mathematics - Numerical Analysis ,0101 mathematics ,Navier–Stokes equations ,Numerical Analysi ,Mathematics ,Pointwise ,Numerical Analysis ,Numerical analysis ,Applied Mathematics ,Mathematical analysis ,Computational mathematics ,Numerical Analysis (math.NA) ,010101 applied mathematics ,Computational Mathematics ,Computational Mathematic ,Element (category theory) ,Polygonal meshe - Abstract
A family of virtual element methods for the two-dimensional Navier-Stokes equations is proposed and analyzed. The schemes provide a discrete velocity field which is pointwise divergence-free. A rigorous error analysis is developed, showing that the methods are stable and optimally convergent. Several numerical tests are presented, confirming the theoretical predictions. A comparison with some mixed finite elements is also performed.
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- 2018
- Full Text
- View/download PDF
12. Virtual element method for second-order elliptic eigenvalue problems
- Author
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Francesca Gardini, Giuseppe Vacca, Gardini, F, and Vacca, G
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Applied Mathematics ,General Mathematics ,010103 numerical & computational mathematics ,16. Peace & justice ,virtual element method, eigenvalue problems, polyhedral meshes ,01 natural sciences ,010101 applied mathematics ,Set (abstract data type) ,Computational Mathematics ,Calculus ,Order (group theory) ,Applied mathematics ,Numerical tests ,0101 mathematics ,Element (category theory) ,Eigenvalues and eigenvectors ,Mathematics - Abstract
We study the virtual element approximation of elliptic eigenvalue problems. The main result of the article states that the virtual element method provides an optimal-order approximation of the eigenmodes. A wide set of numerical tests confirm the theoretical analysis
- Published
- 2018
13. Virtual element methods for parabolic problems on polygonal meshes
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Giuseppe Vacca and Lourenço Beirão da Veiga
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Computational Mathematics ,Numerical Analysis ,Applied Mathematics ,Mathematical analysis ,Polygon mesh ,Element (category theory) ,Analysis ,Mathematics - Published
- 2015
14. Virtual Element Methods for hyperbolic problems on polygonal meshes
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Giuseppe Vacca and Vacca, G
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Mathematical optimization ,Wave propagation ,Virtual Element Method ,Stability (learning theory) ,010103 numerical & computational mathematics ,Volume mesh ,Numerical Analysis (math.NA) ,Wave equation ,01 natural sciences ,010101 applied mathematics ,Hyperbolic problem ,Computational Mathematics ,Computational Theory and Mathematics ,Modeling and Simulation ,Convergence (routing) ,FOS: Mathematics ,Applied mathematics ,Newmark-beta method ,Polygon mesh ,Mathematics - Numerical Analysis ,0101 mathematics ,Element (category theory) ,Mathematics ,Polygonal meshe - Abstract
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.
- Published
- 2016
15. Spectral properties and conservation laws in Mimetic Finite Difference methods for PDEs
- Author
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Giuseppe Vacca, Luciano Lopez, Lopez, L, and Vacca, G
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Conservation law ,Discretization ,Applied Mathematics ,Mathematical analysis ,Linear system ,Runge-Kutta-Chebyshev scheme ,Finite difference ,Finite difference method ,Duality (optimization) ,010103 numerical & computational mathematics ,Differential operator ,01 natural sciences ,Parabolic partial differential equation ,Mathematics::Numerical Analysis ,010101 applied mathematics ,Computational Mathematics ,Mimetic Finite Difference ,0101 mathematics ,Runge–Kutta–Chebyshev scheme ,Mathematics ,Conservation laws - Abstract
The Mimetic Finite Difference (MFD) methods for PDEs mimic crucial properties of mathematical systems: duality and self-adjointness of differential operators, conservation laws and properties of the solution on general polytopal meshes. In this article the structure and the spectral properties of the linear systems derived by the spatial discretization of diffusion problem are analysed. In addition, the numerical approximation of parabolic equations is discussed where the MFD approach is used in the space discretization while implicit ? -method and explicit Runge-Kutta-Chebyshev schemes are used in time discretization. Moreover, we will show how the numerical solution preserves certain conservation laws of the theoretical solution.
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- 2016
16. Divergence free Virtual Elements for the Stokes problem on polygonal meshes
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L. Beirão da Veiga, Giuseppe Vacca, Carlo Lovadina, BEIRAO DA VEIGA, L, Lovadina, C, and Vacca, G
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Virtual space ,010103 numerical & computational mathematics ,Topology ,01 natural sciences ,FOS: Mathematics ,Polygon mesh ,Virtual element method ,Mathematics - Numerical Analysis ,0101 mathematics ,Divergence (statistics) ,Mathematics ,Pointwise ,Numerical Analysis ,Applied Mathematics ,Polygonal meshes ,Degrees of freedom ,Numerical Analysis (math.NA) ,Virtual element method, Polygonal meshes, Stokes Problem, Divergence free approximation ,010101 applied mathematics ,Computational Mathematics ,Stokes problem ,Modeling and Simulation ,Scheme (mathematics) ,Divergence free approximation ,Element (category theory) ,Analysis - Abstract
In the present paper we develop a new family of Virtual Elements for the Stokes problem on polygonal meshes. By a proper choice of the Virtual space of velocities and the associated degrees of freedom, we can guarantee that the final discrete velocity is pointwise divergence-free, and not only in a relaxed (projected) sense, as it happens for more standard elements. Moreover, we show that the discrete problem is immediately equivalent to a reduced problem with fewer degrees of freedom, thus yielding a very efficient scheme. We provide a rigorous error analysis of the method and several numerical tests, including a comparison with a different Virtual Element choice.
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- 2015
17. The Virtual Element Method with curved edges
- Author
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Giuseppe Vacca, A. Russo, L. Beirão da Veiga, Beirão da Veiga, L, Russo, A, and Vacca, G
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Numerical Analysis ,Degree (graph theory) ,Applied Mathematics ,Interface (computing) ,Mathematical analysis ,Boundary (topology) ,010103 numerical & computational mathematics ,Numerical Analysis (math.NA) ,01 natural sciences ,Domain (mathematical analysis) ,010101 applied mathematics ,Computational Mathematics ,Rate of convergence ,Modeling and Simulation ,FOS: Mathematics ,Mathematics - Numerical Analysis ,0101 mathematics ,Element (category theory) ,Virtual element method / polygonal meshes / curved elements ,Virtual element method, polygonal meshes, curved elements ,Analysis ,Mathematics - Abstract
In this paper we initiate the investigation of Virtual Elements with curved faces. We consider the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain or with a curved interface. While an approximation of the domain with polygons leads, for degree of accuracy k≥2, to a sub-optimal rate of convergence, we show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence.
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