Virtual element method for quasilinear elliptic problems

A virtual element method for the quasilinear equation −div(κ(u)gradu)=f using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal-order a priori error estimates in the H1- and L2-norm are proven. In addition, the convergence of fixed-point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes

Hourglass stabilization and the virtual element method

In this paper, we establish the connections between the virtual element method (VEM) and the hourglass control techniques that have been developed since the early 1980s to stabilize underintegrated C0 Lagrange finite element methods. In the VEM, the bilinear form is decomposed into two parts: a consistent term that reproduces a given polynomial space and a correction term that provides stability. The essential ingredients of inline image-continuous VEMs on polygonal and polyhedral meshes are described, which reveals that the variational approach adopted in the VEM affords a generalized and robust means to stabilize underintegrated finite elements. We focus on the heat conduction (Poisson) equation and present a virtual element approach for the isoparametric four-node quadrilateral and eight-node hexahedral elements. In addition, we show quantitative comparisons of the consistency and stabilization matrices in the VEM with those in the hourglass control method of Belytschko and coworkers. Numerical examples in two and three dimensions are presented for different stabilization parameters, which reveals that the method satisfies the patch test and delivers optimal rates of convergence in the L2 norm and the H1 seminorm for Poisson problems on quadrilateral, hexahedral, and arbitrary polygonal meshes

Virtual Element Method for Quasilinear Elliptic Problems

A Virtual Element Method (VEM) for the quasilinear equation −div(κ(u)gradu) = f using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well-posedness of the discrete problem and optimal order a priori error estimates in the H1 - and L2 -norm are proven. In addition, the convergence of fixed point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes.A Virtual Element Method (VEM) for the quasilinear equation −div(κ(u)gradu) = f using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well-posedness of the discrete problem and optimal order a priori error estimates in the H1 - and L2 -norm are proven. In addition, the convergence of fixed point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes

Basic principles of Virtual Element Methods

We present, on the simplest possible case, what we consider as the very basic features of the (brand new) virtual element method. As the readers will easily recognize, the virtual element method could easily be regarded as the ultimate evolution of the mimetic finite differences approach. However, in their last step they became so close to the traditional finite elements that we decided to use a different perspective and a different name. Now the virtual element spaces are just like the usual finite element spaces with the addition of suitable non-polynomial functions. This is far from being a new idea. See for instance the very early approach of E. Wachspress [A Rational Finite Element Basic (Academic Press, 1975)] or the more recent overview of T.-P. Fries and T. Belytschko [The extended/generalized finite element method: An overview of the method and its applications, Int. J. Numer. Methods Engrg.84 (2010) 253\u2013304]. The novelty here is to take the spaces and the degrees of freedom in such a way that the elementary stiffness matrix can be computed without actually computing these non-polynomial functions, but just using the degrees of freedom. In doing that we can easily deal with complicated element geometries and/or higher-order continuity conditions (like C1, C2, etc.). The idea is quite general, and could be applied to a number of different situations and problems. Here however we want to be as clear as possible, and to present the simplest possible case that still gives the flavor of the whole idea

Basic principles of virtual element methods

ABSTRACT Over the past two decades, meshfree methods (MMs) as a numerical tool for solving PDEs have been welldeveloped. In contrast to finite elements, MMs do not require a mesh to construct the basis functions, which are smooth and non-polynomial functions. This feature makes MMs more appealing for the discretization of field variables in problems where, for instance, higher-order smoothness is needed or mesh distortions introduce a limitation for standard finite elements. Nonetheless, Galerkin MMs require background cells to perform the numerical integration of the weak form integrals. Usually, Gauss integration is employed on a background mesh of finite elements. This introduces integration errors that affect the accuracy, convergence and stability of the method. Several authors have tried to overcome the integration issue resulting in integration schemes that only ensure consistency and substantially improve accuracy, but stability is not guaranteed. In this work, a new approach for Galerkin MMs is introduced, which draws on the recently proposed Virtual Element Method [1], to ensure both consistency and stability of the approximate bilinear form. Benchmark examples in two-and three-dimensions will be presented to demonstrate the accuracy, consistency and stability of the method for Poisson and linear elasticity boundary-value problems. REFERENCES [1] L

hp-Version discontinuous Galerkin methods on polygonal and polyhedral meshes

An hp-version interior penalty discontinuous Galerkin method (DGFEM) for the numerical solution of second-order elliptic partial differential equations on general computational meshes consisting of polygonal/polyhedral elements is presented and analyzed. Utilizing a bounding box concept, the method employs elemental polynomial bases of total degree p (P[subscript p]-basis) defined on the physical space, without the need to map from a given reference or canonical frame. This, together with a new specific choice of the interior penalty parameter which allows for face-degeneration, ensures that optimal a priori bounds may be established, for general meshes including polygonal elements with degenerating edges in two dimensions and polyhedral elements with degenerating faces and/or edges in three dimensions. Numerical experiments highlighting the performance of the proposed method are presented. Moreover, the competitiveness of the p-version DGFEM employing a P[subscript p]-basis in comparison to the conforming p-version finite element method on tensor-product elements is studied numerically for a simple test problem