arXiv subjects

Gianmarco Manzini

Publications and source records attributed to Gianmarco Manzini.

At least 37 records · Page 2Linked to original sources

arXiv2021

A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. We numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the $P_1-P_0$ Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.

math.NA↗

arXiv2021

On arbitrarily regular conforming virtual element methods for elliptic partial differential equations

The Virtual Element Method (VEM) is a very effective framework to design numerical approximations with high global regularity to the solutions of elliptic partial differential equations. In this paper, we review the construction of such approximations for an elliptic problem of order $p_1$ using conforming, finite dimensional subspaces of $ H^{p_2}(Ω)$, where $p_1$ and $p_2$ are two integer numbers such that $p_2 \geq p_1 \geq 1$ and $Ω\in R^2$ is the computational domain. An abstract convergence result is presented in a suitably defined energy norm. The space formulation and major aspects such as the choice and unisolvence of the degrees of freedom are discussed, also providing specific examples corresponding to various practical cases of high global regularity. Finally, the construction of the "enhanced" formulation of the virtual element spaces is also discussed in details with a proof that the dimension of the "regular" and "enhanced" spaces is the same and that the virtual element functions in both spaces can be described by the same choice of the degrees of freedom.

math.NA↗

arXiv2021

Polyhedral Mesh Quality Indicator for the Virtual Element Method

We present the design of a mesh quality indicator that can predict the behavior of the Virtual Element Method (VEM) on a given mesh family or finite sequence of polyhedral meshes (dataset). The mesh quality indicator is designed to measure the violation of the mesh regularity assumptions that are normally considered in the convergence analysis. We investigate the behavior of this new mathematical tool on the lowest-order conforming approximation of the three-dimensional Poisson equation. This work also assesses the convergence rate of the VEM when applied to very general polyhedral meshes, including non-convex and skewed three-dimensional elements. Such meshes are created within an original mesh generation framework, which is designed to allow the generation of meshes with very different sizes, connectivity and geometrical properties. The obtained results show a significant correlation between the quality measured a priori by the indicator and the effective performance of the VEM.

math.NA↗

arXiv2021

Extended virtual element method for two-dimensional linear elastic fracture

In this paper, we propose an eXtended Virtual Element Method (X-VEM) for two-dimensional linear elastic fracture. This approach, which is an extension of the standard Virtual Element Method (VEM), facilitates mesh-independent modeling of crack discontinuities and elastic crack-tip singularities on general polygonal meshes. For elastic fracture in the X-VEM, the standard virtual element space is augmented by additional basis functions that are constructed by multiplying standard virtual basis functions by suitable enrichment fields, such as asymptotic mixed-mode crack-tip solutions. The design of the X-VEM requires an extended projector that maps functions lying in the extended virtual element space onto a set spanned by linear polynomials and the enrichment fields. An efficient scheme to compute the mixed-mode stress intensity factors using the domain form of the interaction integral is described. The formulation permits integration of weakly singular functions to be performed over the boundary edges of the element. Numerical experiments are conducted on benchmark mixed-mode linear elastic fracture problems that demonstrate the sound accuracy and optimal convergence in energy of the proposed formulation.

math.NA↗

arXiv2021

Virtual element approximation of two-dimensional parabolic variational inequalities

We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element method is based on the lowest-order virtual element space that contains the subspace of the linear polynomials defined on each element. The connection between the nonnegativity of the virtual element functions and the nonnegativity of the degrees of freedom, i.e., the values at the mesh vertices, is established by applying the Maximum and Minimum Principle Theorem. The mass matrix is computed through an approximate L 2 polynomial projection, whose properties are carefully investigated in the paper. We prove the well-posedness of the resulting scheme in two different ways that reveal the contractive nature of the VEM and its connection with the minimization of quadratic functionals. The convergence analysis requires the existence of a nonnegative quasi-interpolation operator, whose construction is also discussed in the paper. The variational crime introduced by the virtual element setting produces five error terms that we control by estimating a suitable upper bound. Numerical experiments confirm the theoretical convergence rate for the refinement in space and time on three different mesh families including distorted squares, nonconvex elements, and Voronoi tesselations.

math.NA↗

arXiv2021

Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

We develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

math.NA↗

arXiv2021

The virtual element method for the coupled system of magneto-hydrodynamics

In this work, we review the framework of the Virtual Element Method (VEM) for a model in magneto-hydrodynamics (MHD), that incorporates a coupling between electromagnetics and fluid flow, and allows us to construct novel discretizations for simulating realistic phenomenon in MHD. First, we study two chains of spaces approximating the electromagnetic and fluid flow components of the model. Then, we show that this VEM approximation will yield divergence free discrete magnetic fields, an important property in any simulation in MHD. We present a linearization strategy to solve the VEM approximation which respects the divergence free condition on the magnetic field. This linearization will require that, at each non-linear iteration, a linear system be solved. We study these linear systems and show that they represent well-posed saddle point problems. We conclude by presenting numerical experiments exploring the performance of the VEM applied to the subsystem describing the electromagnetics. The first set of experiments provide evidence regarding the speed of convergence of the method as well as the divergence-free condition on the magnetic field. In the second set we present a model for magnetic reconnection in a mesh that includes a series of hanging nodes, which we use to calibrate the resolution of the method. The magnetic reconnection phenomenon happens near the center of the domain where the mesh resolution is finer and high resolution is achieved.

math.NA↗

arXiv2021

A review on arbitrarily regular conforming virtual element methods for elliptic partial differential equations

The Virtual Element Method is well suited to the formulation of arbitrarily regular Galerkin approximations of elliptic partial differential equations of order $2p_1$, for any integer $p_1\geq 1$. In fact, the virtual element paradigm provides a very effective design framework for conforming, finite dimensional subspaces of $H^{p_2}(Ω)$, $Ω$ being the computational domain and $p_2\geq p_1$ another suitable integer number. In this study, we first present an abstract setting for such highly regular approximations and discuss the mathematical details of how we can build conforming approximation spaces with a global high-order continuity on $Ω$. Then, we illustrate specific examples in the case of second- and fourth-order partial differential equations, that correspond to the cases $p_1=1$ and $2$, respectively. Finally, we investigate numerically the effect on the approximation properties of the conforming highly-regular method that results from different choices of the degree of continuity of the underlying virtual element spaces and how different stabilization strategies may impact on convergence.

math.NA↗

arXiv2021

A decision-making machine learning approach in Hermite spectral approximations of partial differential equations

The accuracy and effectiveness of Hermite spectral methods for the numerical discretization of partial differential equations on unbounded domains, are strongly affected by the amplitude of the Gaussian weight function employed to describe the approximation space. This is particularly true if the problem is under-resolved, i.e., there are no enough degrees of freedom. The issue becomes even more crucial when the equation under study is time-dependent, forcing in this way the choice of Hermite functions where the corresponding weight depends on time. In order to adapt dynamically the approximation space, it is here proposed an automatic decision-making process that relies on machine learning techniques, such as deep neural networks and support vector machines. The algorithm is numerically tested with success on a simple 1D problem, but the main goal is its exportability in the context of more serious applications.

math.NA↗

arXiv2021

Stabilization of the nonconforming virtual element method

We address the issue of designing robust stabilization terms for the nonconforming virtual element method. To this end, we transfer the problem of defining the stabilizing bilinear form from the elemental nonconforming virtual element space, whose functions are not known in closed form, to the dual space spanned by the known functionals providing the degrees of freedom. By this approach, we manage to construct different bilinear forms yielding optimal or quasi-optimal stability bounds and error estimates, under weaker assumptions on the tessellation than the ones usually considered in this framework. In particular, we prove optimality under geometrical assumptions allowing a mesh to have a very large number of arbitrarily small edges per element. Finally, we numerically assess the performance of the VEM for several different stabilizations fitting with our new framework on a set of representative test cases.

math.NA↗

arXiv2021

VEM and the Mesh

In this work we report some results, obtained within the framework of the ERC Project CHANGE, on the impact on the performance of the virtual element method of the shape of the polygonal elements of the underlying mesh. More in detail, after reviewing the state of the art, we present a) an experimental analysis of the convergence of the VEM under condition violating the standard shape regularity assumptions, b) an analysis of the correlation between some mesh quality metrics and a set of different performance indexes, and c) a suitably designed mesh quality indicator, aimed at predicting the quality of the performance of the VEM on a given mesh.

math.NA↗

arXiv2021

Stability and conservation properties of Hermite-based approximations of the Vlasov-Poisson system

Spectral approximation based on Hermite-Fourier expansion of the Vlasov-Poisson model for a collisionless plasma in the electro-static limit is provided, by including high-order artificial collision operators of Lenard-Bernstein type. These differential operators are suitably designed in order to preserve the physically-meaningful invariants (number of particles, momentum, energy). In view of time-discretization, stability results in appropriate norms are presented. In this study, necessary conditions link the magnitude of the artificial collision term, the number of spectral modes of the discretization, as well as the time-step. The analysis, carried out in full for the Hermite discretization of a simple linear problem in one-dimension, is then partly extended to cover the complete nonlinear Vlasov-Poisson model.

math.NA↗

arXiv2021

The role of mesh quality and mesh quality indicators in the Virtual Element Method

Since its introduction, the Virtual Element Method (VEM) was shown to be able to deal with a large variety of polygons, while achieving good convergence rates. The regularity assumptions proposed in the VEM literature to guarantee the convergence on a theoretical basis are therefore quite general. They have been deduced in analogy to the similar conditions developed in the Finite Element Methods (FEMs) analysis. In this work, we experimentally show that the VEM still converges with almost optimal rates and low errors in the L2 and H1 norms even if we significantly break the regularity assumptions that are used in the literature. These results suggest that the regularity assumptions proposed so far might be overestimated. We also exhibit examples on which the VEM sub-optimally converges or diverges. Finally, we introduce a mesh quality indicator that experimentally correlates the entity of the violation of the regularity assumptions and the performance of the VEM solution, thus predicting if a dataset is potentially critical for VEM.

math.NA↗

arXiv2021

The conforming virtual element method for polyharmonic and elastodynamics problems: a review

In this paper, we review recent results on the conforming virtual element approximation of polyharmonic and elastodynamics problems. The structure and the content of this review is motivated by three paradigmatic examples of applications: classical and anisotropic Cahn-Hilliard equation and phase field models for brittle fracture, that are briefly discussed in the first part of the paper. We present and discuss the mathematical details of the conforming virtual element approximation of linear polyharmonic problems, the classical Cahn-Hilliard equation and linear elastodynamics problems.

math.NA↗

arXiv2020

The multi-dimensional Hermite-discontinuous Galerkin method for the Vlasov-Maxwell equations

We discuss the development, analysis, implementation, and numerical assessment of a spectral method for the numerical simulation of the three-dimensional Vlasov-Maxwell equations. The method is based on a spectral expansion of the velocity space with the asymmetrically weighted Hermite functions. The resulting system of time-dependent nonlinear equations is discretized by the discontinuous Galerkin (DG) method in space and by the method of lines for the time integration using explicit Runge-Kutta integrators. The resulting code, called Spectral Plasma Solver (SPS-DG), is successfully applied to standard plasma physics benchmarks to demonstrate its accuracy, robustness, and parallel scalability.

physics.comp-ph↗

arXiv2018

A Semi-Lagrangian Spectral Method for the Vlasov-Poisson System based on Fourier, Legendre and Hermite Polynomials

In this work, we apply a semi-Lagrangian spectral method for the Vlasov-Poisson system, previously designed for periodic Fourier discretizations, by implementing Legendre polynomials and Hermite functions in the approximation of the distribution function with respect to the velocity variable. We discuss second-order accurate-in-time schemes, obtained by coupling spectral techniques in the space-velocity domain with a BDF time-stepping scheme. The resulting method possesses good conservation properties, which have been assessed by a series of numerical tests conducted on the standard two-stream instability benchmark problem. In the Hermite case, we also investigate the numerical behavior in dependence of a scaling parameter in the Gaussian weight. Confirming previous results from the literature, our experiments for different representative values of this parameter, indicate that a proper choice may significantly impact on accuracy, thus suggesting that suitable strategies should be developed to automatically update the parameter during the time-advancing procedure.

math.NA↗

arXiv2018

SUPG stabilization for the nonconforming virtual element method for advection-diffusion-reaction equations

We present the design, convergence analysis and numerical investigations of the nonconforming virtual element method with Streamline Upwind/Petrov-Galerkin (VEM-SUPG) stabilization for the numerical resolution of convection-diffusion-reaction problems in the convective-dominated regime. According to the virtual discretization approach, the bilinear form is split as the sum of a consistency and a stability term. The consistency term is given by substituting the functions of the virtual space and their gradients with their polynomial projection in each term of the bilinear form (including the SUPG stabilization term). Polynomial projections can be computed exactly from the degrees of freedom. The stability term is also built from the degrees of freedom by ensuring the correct scalability properties with respect to the mesh size and the equation coefficients. The nonconforming formulation relaxes the continuity conditions at cell interfaces and a weaker regularity condition is considered involving polynomial moments of the solution jumps at cell interface. Optimal convergence properties of the method are proved in a suitable norm, which includes a contribution from the advective stabilization terms. Experimental results confirm the theoretical convergence rates.

math.NA↗

arXiv2018

Arbitrary-order time-accurate semi-Lagrangian spectral approximations of the Vlasov-Poisson system

The Vlasov-Poisson system, modeling the evolution of non-collisional plasmas in the electrostatic limit, is approx- imated by a Semi-Lagrangian technique. Spectral methods of periodic type are implemented through a collocation approach. Groups of particles are represented by the Fourier Lagrangian basis and evolve, for a single timestep, along an high-order accurate representation of the local characteristic lines. The time-advancing technique is based on Taylor developments that can be, in principle, of any order of accuracy, or by coupling the phase space discretiza- tion with high-order accurate Backward Differentiation Formulas (BDF) as in the method-of-lines framework. At each timestep, particle displacements are reinterpolated and expressed in the original basis to guarantee the order of accuracy in all the variables at relatively low costs. Thus, these techniques combine excellent features of spectral approximations with high-order time integration. Series of numerical experiments are performed in order to assess the real performance. In particular, comparisons with standard benchmarks are examined.

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