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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 that confirm the theoretical error bounds are also presented.

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.

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.

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.

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.

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

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.

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.

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.

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.

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

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.

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.

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.

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.