New perspectives on polygonal and polyhedral finite element methods

Abstract

Generalized barycentric coordinates such as Wachspress and mean value coordinates have been used in polygonal and polyhedral finite element methods. Recently, mimetic finite difference schemes were cast within a variational framework, and a consistent and stable finite element method on arbitrary polygonal meshes was devised. The method was coined as the virtual element method (VEM), since it did not require the explicit construction of basis functions. This advance provides a more in-depth understanding of mimetic schemes, and also endows polygonal-based Galerkin methods with greater flexibility than three-node and four-node finite element methods. In the VEM, a projection operator is used to realize the decomposition of the stiffness matrix into two terms: a consistent matrix that is known, and a stability matrix that must be positive semi-definite and which is only required to scale like the consistent matrix. In this paper, we first present an overview of previous developments on conforming polygonal and polyhedral finite elements, and then appeal to the exact decomposition in the VEM to obtain a robust and efficient generalized barycentric coordinate-based Galerkin method on polygonal and polyhedral elements. The consistent matrix of the VEM is adopted, and numerical quadrature with generalized barycentric coordinates is used to compute the stability matrix. This facilitates post-processing of field variables and visualization in the VEM, and on the other hand, provides a means to exactly satisfy the patch test with efficient numerical integration in polygonal and polyhedral finite elements. We present numerical examples that demonstrate the sound accuracy and performance of the proposed method. For Poisson problems in ℝ2 and ℝ3, we establish that linearly complete generalized barycentric interpolants deliver optimal rates of convergence in the L2-norm and the H1-seminorm.

Novel Polyhedral Finite Elements for Numerical Analysis

“…It is seen that the range for the error norms of the VPHE elements are similar to those obtained using Wachspress and mean value coordinate for 2-dimensions as reported in [3] and 3D PFEM as reported in [4] for 3-dimensions. It is also observed that the error norm in heat transfer phenomena is lower.…”

Virtual Elements for the Navier--Stokes Problem on Polygonal Meshes

“…With that choice, a theoretical analysis can be developed using the guidelines in [41] in connection with the same tools and ideas of section 4. Here we prefer to consider the choice (42), which allows for a more direct stability argument. Nevertheless, in the numerical tests of section 5 we will investigate both possibilities.…”

“…Among the Galerkin schemes, VEM is peculiar in that the discrete spaces consist of functions which are not known pointwise, but about which a limited set of information is available. This limited information is sufficient to construct the stiffness matrix and the right-hand side.The VEM has been developed for many problems; see, for example, [23,1,10,43,46,9,19,17,18,6,42,54,51,37,45]. More specifically, with regard to the Stokes problem, virtual elements have been developed in [3,28,15,25,26,52].…”

The virtual element method in 50 lines of MATLAB

“…These local problems are designed in such a way that the virtual element space includes a subspace of polynomials of some prescribed degree (referred to as the degree of the method) alongside other, typically unknown, virtual functions. In this respect, and like many other conforming approaches to polygonal meshes such as the polygonal finite element method [18,27] or BEM-based FEM [22], the virtual element method falls within the broad class of generalised finite element methods [25]. What sets the virtual element method apart from these other approaches, however, is that the extra non-polynomial virtual functions never need to be determined or evaluated in practice.…”