Virtual element method for semilinear elliptic problems on polygonal meshes

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.

Two-grid virtual element discretization of semilinear elliptic problem

Virtual element discretization method to optimal control problem governed by Stokes equations with pointwise control constraint on arbitrary polygonal meshes

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

A framework for developing a mimetic tensor artificial viscosity for Lagrangian hydrocodes on arbitrary polygonal meshes

A two-grid method for semi-linear elliptic interface problems by partially penalized immersed finite element methods

With the rising complexity of architectural design, a growing interest in topology optimization, and the increasing need to model crack propagation, there is a higher demand for highly flexible geometry descriptions that go hand in hand with robust element formulations. Methods relying on polygonal meshes, such as the virtual element method (VEM), element formulations based on Wachspress shape functions, and the scaled boundary finite element method (SBFEM) have proven to be versatile in solving physical problems. Arbitrarily shaped elements are beneficial when it comes to highly localized meshes, treatment of hanging nodes, and flexibility of the meshing of complex domains. This paper provides a comparative study of polygonal element formulations, focusing specifically on the VEM, the semi‐analytical SBFEM, and the fully discretized SBFEM. The study evaluates these methods in terms of stability, convergence behavior, and flexibility in the context of linear elasticity. The potential of the formulations is evaluated by applying them to several benchmark problems and by comparing the results. In doing so, the flexibility of the considered formulations is also proven. To this end, several indicators are considered, such as error norms, convergence rates, and displacements.

Convergence analysis of a new dynamic diffusion method

Neumann problems of 2D Laplace's equation by method of fundamental solutions

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.

We present a finite volume scheme for 2D steady convection–diffusion equations on arbitrary convex polygonal meshes. The method uses auxiliary unknowns on the cell‐edge and cell‐node of the mesh, and the continuous flux is computed to be a two‐point nonlinear flux. Since the transpose of the coefficient matrix is an M‐matrix, it is guaranteed that the scheme is positive. We propose a new strategy, which aims that providing a scheme without separated method to compute the value on the cell‐edge without previous reconstruct. Numerical results show that our scheme is positive on polygonal meshes for both diffusion‐dominated and convection‐dominated problems and has second‐order accuracy for solutions.

This paper extends the applicability of the combined use of the virtual element method (VEM) and the boundary element method (BEM), recently introduced to solve the coupling of linear elliptic equations in divergence form with the Laplace equation, to the case of acoustic scattering problems in 2D and 3D. The well-posedness of the continuous and discrete formulations are established, and then Cea-type estimates and consequent rates of convergence are derived.

The virtual element method (VEM) is a recent technology that can make use of very general polygonal/polyhedral meshes without the need to integrate complex nonpolynomial functions on the elements and preserving an optimal order of convergence. In this article, we develop for the first time, the VEM for parabolic problems on polygonal meshes, considering time‐dependent diffusion as our model problem. After presenting the scheme, we develop a theoretical analysis and show the practical behavior of the proposed method through a large array of numerical tests. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 2110–2134, 2015

Superconvergence analysis of an [formula omitted]-Galerkin mixed FEM for Klein–Gordon–Zakharov equations with power law nonlinearity