We analyse the interpolation properties of 2D and 3D low order virtual element face and edge spaces, which generalize Nédélec and Raviart-Thomas polynomials to polygonal-polyhedral meshes. Moreover, we investigate the stability properties of the associated $L^2$ discrete bilinear forms, which typically appear in the virtual element discretization of problems in electromagnetism.
We develop interpolation error estimates for general order standard and serendipity edge and face virtual elements in two and three dimensions. Contextually, we investigate the stability properties of the associated L2 discrete bilinear forms. These results are fundamental tools in the analysis of general order virtual elements, e.g., for electromagnetic problems.
We prove stability bounds for Stokes-like virtual element spaces in two and three dimensions. Such bounds are also instrumental in deriving optimal interpolation estimates. Furthermore, we develop some numerical tests in order to investigate the behaviour of the stability constants also from the practical side.
Stabilization-free virtual element methods based on finite element interpolation
Zhengzhou University（郑州大学）；Henan University of Technology（河南工业大学）；Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；University of Chinese Academy of Sciences（中国科学院大学）
In this paper, we employ the techniques developed for second order operators to obtain the new estimates of Virtual Element Method for fourth order operators. The analysis is based on elements with proper shape regularity. Estimates for projection and interpolation operators are derived. Also, the biharmonic problem is solved by Virtual Element Method, optimal error estimates are obtained. Our choice of the discrete form for the right hand side function relaxes the regularity requirement in previous work and the error estimates between exact solutions and the computable numerical solutions are…
A unified construction of the $H^m$-nonconforming virtual elements of any order $k$ is developed on any shape of polytope in $\mathbb R^n$ with constraints $m\leq n$ and $k\geq m$. As a vital tool in the construction, a generalized Green's identity for $H^m$ inner product is derived. The $H^m$-nonconforming virtual element methods are then used to approximate solutions of the $m$-harmonic equation. After establishing a bound on the jump related to the weak continuity, the optimal error estimate of the canonical interpolation, and the norm equivalence of the stabilization term, the optimal erro…
Piecewise divergence-free nonconforming virtual elements are designed for Stokes problem in any dimensions. After introducing a local energy projector based on the Stokes problem and the stabilization, a divergence-free nonconforming virtual element method is proposed for Stokes problem. A detailed and rigorous error analysis is presented for the discrete method. An important property in the analysis is that the local energy projector commutes with the divergence operator. With the help of a divergence-free interpolation operator onto a generalized Raviart-Thomas element space, a pressure-robu…
The purpose of the present paper is to develop $C^1$ Virtual Elements in three dimensions for linear elliptic fourth order problems, motivated by the difficulties that standard conforming Finite Elements encounter in this framework. We focus the presentation on the lowest order case, the generalization to higher orders being briefly provided in the Appendix. The degrees of freedom of the proposed scheme are only 4 per mesh vertex, representing function values and gradient values. Interpolation error estimates for the proposed space are provided, together with a set of numerical tests to valida…
In the present work we generalize the curvilinear Virtual Element technology, introduced for a simple linear scalar problem in a previous work, to generic 2D solid mechanic problems in small deformations. Such generalization also includes the development of a novel Virtual Element space for displacements that contains rigid body motions. Our approach can accept a generic black-box (elastic or inelastic) constitutive algorithm and, in addition, can make use of curved edges thus leading to an exact approximation of the geometry. Rigorous theoretical interpolation properties for the new space on …
The $H^m$-nonconforming virtual elements of any order $k$ on any shape of polytope in $\mathbb R^n$ with constraints $m>n$ and $k\geq m$ are constructed in a universal way. A generalized Green's identity for $H^m$ inner product with $m>n$ is derived, which is essential to devise the $H^m$-nonconforming virtual elements. By means of the local $H^m$ projection and a stabilization term using only the boundary degrees of freedom, the $H^m$-nonconforming virtual element methods are proposed to approximate solutions of the $m$-harmonic equation. The norm equivalence of the stabilization on the kerne…
We present a low order virtual element discretization for time dependent Maxwell's equations, which allow for the use of general polyhedral meshes. Both the semi- and fully-discrete schemes are considered. We derive optimal a priori estimates and validate them on a set of numerical experiments. As pivot results, we discuss some novel inequalities for de Rahm sequences of nodal, edge, and face virtual element spaces.
We analyse the p- and hp-versions of the virtual element method (VEM) for the the Stokes problem on a polygonal domain. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from [7] an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accu…
We design a virtual element method for the numerical treatment of the two-dimensional parabolic variational inequality problem on unstructured polygonal meshes. Due to the expected low regularity of the exact solution, the virtual element method is based on the lowest-order virtual element space that contains the subspace of the linear polynomials defined on each element. The connection between the nonnegativity of the virtual element functions and the nonnegativity of the degrees of freedom, i.e., the values at the mesh vertices, is established by applying the Maximum and Minimum Principle Th…
The lowest-order nonconforming virtual element extends the Morley triangular element to polygons for the approximation of the weak solution $u\in V:=H^2_0(Ω)$ to the biharmonic equation. The abstract framework allows (even a mixture of) two examples of the local discrete spaces $V_h(P)$ and a smoother allows rough source terms $F\in V^*=H^{-2}(Ω)$. The a priori and a posteriori error analysis in this paper circumvents any trace of second derivatives by some computable conforming companion operator $J:V_h\to V$ from the nonconforming virtual element space $V_h$. The operator $J$ is a right-inve…
Non divergence-free discretisations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterised by large discretisations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods (VEM) on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretised in a careful manner. To be able to evaluate the right-hand side for the test functions, some explicit interpolation of the virtual test functions is needed that can be evaluated pointwise everywhere. The sta…
