We present, on the simplest possible case, what we consider as very basic features of (brand new) virtual element method. As readers will easily recognize, method could be regarded ultimate evolution mimetic finite differences approach. However, in their last step they became so close to traditional elements that decided use a different perspective and name. Now spaces are just like usual with addition suitable non-polynomial functions. This is far from being new idea. See for instance early...

We introduce the nonconforming Virtual Element Method (VEM) for approximation of second order elliptic problems. present construction new element in two and three dimensions, highlighting main differences with conforming VEM classical finite methods. provide error analysis establish equivalence a family mimetic difference Numerical experiments verify theory validate performance proposed method.

We present, in a unified framework, new conforming and nonconforming virtual element methods for general second-order elliptic problems two three dimensions. The differential operator is split into its symmetric nonsymmetric parts conditions stability accuracy on their discrete counterparts are established. These shown to lead optimal |$H^1$|- |$L^2$|-error estimates, confirmed by numerical experiments set of polygonal meshes. the approximation provided be comparable.

In this article, we analyze a discontinuous finite element method recently introduced by Bassi and Rebay for the approximation of elliptic problems. Stability error estimates in various norms are proven. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 365–378,

We develop and analyse a new family of virtual element methods on unstructured polygonal meshes for the diffusion problem in primal form, which uses arbitrarily regular discrete spaces Vh ⊂ Cα, α ∈ ℕ. The degrees freedom are (a) solution derivative values various at suitable nodes (b) moments inside polygons. convergence method is proved theoretically an optimal error estimate derived. Numerical experiments confirm rate that expected from theory.

Generalized barycentric coordinates such as Wachspress and mean value have been used in polygonal polyhedral finite element methods. Recently, mimetic difference schemes were cast within a variational framework, consistent stable method on arbitrary meshes was devised. The coined the virtual (VEM), since it did not require explicit construction of basis functions. This advance provides more in-depth understanding schemes, also endows polygonal-based Galerkin methods with greater flexibility...

We present the nonconforming virtual element method (VEM) for numerical approximation of velocity and pressure in steady Stokes problem. The is approximated using discontinuous piecewise polynomials, while each component space. On mesh local space contains polynomials up to a given degree, plus suitable nonpolynomial functions. functions are implicitly defined as solution Poisson problems with polynomial Neumann boundary conditions. As typical VEM approaches, explicit evaluation...

In this paper, we address the numerical approximation of linear fourth-order elliptic problems on polygonal meshes. particular, present a novel nonconforming virtual element discretization arbitrary order accuracy for biharmonic problems. The space is made possibly discontinuous functions, thus giving rise to fully method. We derive optimal error estimates in suitable (broken) energy norm and results assess validity theoretical estimates.

We develop and analyze a new family of mimetic methods on unstructured polygonal meshes for the diffusion problem in primal form. These are derived from local consistency condition that is exact polynomials any degree $m\geq1$. The degrees freedom (a) solution values at quadrature nodes Gauss–Lobatto formulas each mesh edge, (b) moments inside polygons. convergence method proven theoretically an optimal error estimate mesh-dependent norm mimics energy norm. Numerical experiments confirm rate...

A posteriori error estimation and adaptivity are very useful in the context of virtual element mimetic discretization methods due to flexibility meshes which these numerical schemes can be applied. Nevertheless, developing estimators for is not a straightforward task lack knowledge basis functions. In new setting, we develop residual based estimator Poisson problem with (piecewise) constant coefficients, that proven reliable efficient. We moreover show performance proposed when it combined...

Summary In this paper, we establish the connections between virtual element method (VEM) and hourglass control techniques that have been developed since early 1980s to stabilize underintegrated C 0 Lagrange finite methods. VEM, bilinear form is decomposed into two parts: a consistent term reproduces given polynomial space correction provides stability. The essential ingredients of ‐continuous VEMs on polygonal polyhedral meshes are described, which reveals variational approach adopted in VEM...

We analyse the nonconforming Virtual Element Method (VEM) for approximation of elliptic eigenvalue problems. The VEM allows to treat in same formulation two- and three-dimensional case. present two possible formulations discrete problem, derived respectively by nonstabilized stabilized L 2 -inner product, we study convergence properties corresponding problem. proposed schemes provide a correct spectrum, particular prove optimal-order error estimates eigenfunctions usual double order...

Abstract We analyze the joint efforts made by geometry processing and numerical analysis communities in last decades to define measure concept of “mesh quality”. Researchers have been striving determine how, how much, accuracy a simulation or scientific computation (e.g., rendering, printing, modeling operations) depends on particular mesh adopted model problem, which geometrical features most influence result. The goal was produce with good properties lowest possible number elements, able...

A cell-centered finite volume method is proposed to approximate numerically the solution steady convection-diffusion equation on unstructured meshes of d-simplexes, where $d\geq 2$ spatial dimension. The formally second-order accurate by means a piecewise linear reconstruction within each cell and at mesh vertices. An algorithm provided calculate nonnegative bounded weights. Face gradients, required discretize diffusive fluxes, are defined nonlinear strategy that allows us demonstrate...

In this paper we extend the discrete duality finite volume (DDFV) formulation to steady convection-diffusion equation. The gradients defined in DDFV are used define a cell-based gradient for control volumes of both primal and dual meshes, order achieve higher-order accurate numerical flux convection term. A priori analysis is carried out show convergence approximation, global first-order rate derived. theoretical results confirmed by some experiments.