Nonconforming virtual element methods for velocity-pressure Stokes eigenvalue problem

Convergence analysis of virtual element method for the electric interface model on polygonal meshes with small edges

Conforming virtual element approximations of the two-dimensional Stokes problem

We present the details of the implementation of the Virtual Element Method (VEM) for nonlinear elliptic and parabolic problems, over polygonal meshes. The VEM implementation of a nonlinear problem does not follow the traditional way of implementation of the nonlinear problems as done in finite element/finite volume or finite difference methods. To facilitate this, suitable projection operators are used for the nonlinear terms that are rewritten in terms of the degrees of freedom associated with the virtual element space and thus are computable. We explicitly present the discretization and include sample codes that will help readers with the basic implementation of the nonlinear problems.

The stabilized nonconforming virtual element method for the Darcy–Stokes problem

This paper presents stochastic virtual element methods for propagating uncertainty in linear elastic stochastic problems. We first derive stochastic virtual element equations for 2D and 3D linear elastic problems that may involve uncertainties in material properties, external forces, boundary conditions, etc. A stochastic virtual element space that couples the deterministic virtual element space and the stochastic space is constructed for this purpose and used to approximate the unknown stochastic solution. Two numerical frameworks are then developed to solve the derived stochastic virtual element equations, including a Polynomial Chaos approximation based approach and a weakly intrusive approximation based approach. In the Polynomial Chaos based framework, the stochastic solution is approximated using the Polynomial Chaos basis and solved via an augmented deterministic virtual element equation that is generated by applying the stochastic Galerkin procedure to the original stochastic virtual element equation. In the weakly intrusive approximation based framework, the stochastic solution is approximated by a summation of a set of products of random variables and deterministic vectors, where the deterministic vectors are solved via converting the original stochastic problem to deterministic virtual element equations by the stochastic Galerkin approach, and the random variables are solved via converting the original stochastic problem to one-dimensional stochastic algebraic equations by the classical Galerkin procedure. This method avoids the curse of dimensionality in high-dimensional stochastic problems successfully since all random inputs are embedded into one-dimensional stochastic algebraic equations whose computational effort weakly depends on the stochastic dimension. Numerical results on 2D and 3D problems with low- and high-dimensional random inputs demonstrate the good performance of the proposed methods.

Conforming and nonconforming conservative virtual element methods for nonlinear Schrödinger equation: A unified framework

• Weak formulation of configurational forces (CFs) in the VEM is derived. • Automated mesh refinement based on CFs improves VEM approximations. • Accurate crack tip loading analyses are demonstrated. • Internal moments in virtual elements must not be disregarded in fracture analyses. • Efficient and accurate crack growth simulations with non-convex crack tip elements. The Virtual Element Method (VEM) constitutes a generalization of the Finite Element Method, allowing for spatial discretizations with arbitrary, even non-convex polygonal elements. This offers significant advantages, particularly for problems involving moving discontinuities, local stress concentrations, or multiple phases with distinctly different dimensions. The theory of configurational forces (CF) represents a very flexible basis to quantify driving forces for virtual defect displacements within a thermodynamical framework, and at the same time provides an indicator for the local mesh quality in numerical calculations. In combination, the VEM and the CF theory thus constitute an effective basis for numerical simulations of crack growth within the context of a classical approach with strong discontinuities and sharp crack tips. Challenges to cope with are attributable to the lacking knowledge of shape functions within virtual elements, the numerical integration on arbitrary polygonal elements, and the requirement of an automated generation of polygonal meshes. The weak form of the energy-momentum balance is derived from a variational approach, uniquely separating nodal CFs acting at free surfaces from those driving the defect. Furthermore, internal degrees of freedom, which are inherent to higher-order virtual elements, have to be interpreted within the context of CFs. Unphysical, so-called spurious, CFs are exploited for an improved mesh design at crack tips, eventually generating accurate results of crack tip loading and crack path simulation.

SummaryIn 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 ‐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. Copyright © 2015 John Wiley & Sons, Ltd.

We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.

Finite element methods for electromagnetic problems modeled by Maxwell-type equations are highly sensitive to the conformity of approximation spaces, and non-conforming methods may cause loss of convergence. This fact leads to an essential obstacle for almost all the interface-unfitted mesh methods in the literature regarding the application to electromagnetic interface problems, as they are based on non-conforming spaces. In this work, a novel immersed virtual element method for solving a three-dimensional (3D) H(curl) interface problem is developed, and the motivation is to combine the conformity of virtual element spaces and robust approximation capabilities of immersed finite element spaces. The proposed method is able to achieve optimal convergence. To develop a systematic framework, the [Formula: see text], H(curl) and H(div) interface problems and their corresponding problem-orientated immersed virtual element spaces are considered all together. In addition, the de Rham complex will be established based on which the Hiptmair–Xu (HX) preconditioner can be used to develop a fast solver for the H(curl) interface problem.

In this work, we present an abstract error analysis framework for the approximation of linear partial differential equation (PDE) problems in weak formulation. We consider approximation methods in fully discrete formulation, where the discrete and continuous spaces are possibly not embedded in a common space. A proper notion of consistency is designed, and, under a classical inf-sup condition, it is shown to bound the approximation error. This error estimate result is in the spirit of Strang's first and second lemmas, but applicable in situations not covered by these lemmas (because of a fully discrete approximation space). An improved estimate is also established in a weaker norm, using the Aubin--Nitsche trick. We then apply these abstract estimates to an anisotropic heterogeneous diffusion model and two classical families of schemes for this model: Virtual Element and Finite Volume methods. For each of these methods, we show that the abstract results yield new error estimates with a precise and mild dependency on the local anisotropy ratio. A key intermediate step to derive such estimates for Virtual Element Methods is proving optimal approximation properties of the oblique elliptic projector in weighted Sobolev seminorms. This is a result whose interest goes beyond the specific model and methods considered here. We also obtain, to our knowledge, the first clear notion of consistency for Finite Volume methods, which leads to a generic error estimate involving the fluxes and valid for a wide range of Finite Volume schemes. An important application is the first error estimate for Multi-Point Flux Approximation L and G methods. In the appendix, not included in the published version of this work, we show that classical estimates for discontinuous Galerkin methods can be obtained with simplified arguments using the abstract framework.

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 }) and prove that the triad {{varPhi }_h, {varvec{V}}_h, Q_h} (with {varvec{V}}_h and 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.

This paper proposes virtual element methods for approximating the mathematical model consisting of coupled poroelastic and Advection-Diffusion-Reaction (ADR) equations. The space discretization relies on virtual element spaces containing piecewise linear polynomials as well as non-polynomials for displacement, pressure and concentrations, and piecewise constant for total pressure; a backwardEuler scheme is employed for the approximation of time derivative. Using standard techniques of explicit schemes, the well-posedness of the resultant fully discrete scheme is derived. Moreover, under certain regularity assumptions on the mesh, optimal apriori error estimates are established by introducing suitable projection operators. Several numerical experiments are presented to validate the theoretical convergence rate and exhibit the proposed scheme’s performance.

A variationally consistent numerical approach based on the Virtual Element Method (VEM) is presented for the analysis of 2D elastoplasticity problems. The mixed Hu-Washizu functional of elasticity is extended to incorporate the energy contributions specific to the finite-step elastoplastic problem. It is demonstrated how the governing equations of the discretized elastoplastic problem - including the loading-unloading conditions - emerge naturally as the stationarity conditions of the VEM-discretized functional. Spurious hourglass modes are prevented by formulating a self-stabilized version of Virtual Elements (VEs) that exploits the possibility offered by the mixed approach to define strain and displacement approximations of the same order. The insensitivity of VEs to element distortion and the possibility to use polygonal elements with any shape and number of edges is tested with the analysis of several benchmarks from the literature. It is shown how accurate solutions can be obtained also in the case of non-convex quadrilateral or pentagonal elements. Additionally, the role of internal moment degrees of freedom in preventing elastoplastic locking at the plastic failure limit is elucidated.

Bricks for the mixed high-order virtual element method: Projectors and differential operators