Daniele Boffi, Francesca Gardini, Lucia Gastaldi
We discuss the approximation of eigenvalue problems associated with elliptic partial differential equations using the virtual element method. After recalling the abstract theory, we present a model problem, describing in detail the features of the scheme, and highligting the effects of the stabilizing parameters. We conlcude the discussion with a survey of several application examples.
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Virtual Element Method for Second Order Elliptic Eigenvalue Problems
本文提出虚拟元方法用于二阶椭圆特征值问题，证明该方法能提供最优阶近似，并通过大量数值实验验证了理论分析。
A Conforming virtual element approximation for the Oseen eigenvalue problem
本文提出并分析了一种协调虚拟元方法，用于求解二维Oseen特征值问题，通过紧算子理论推导了先验误差估计，并用数值试验验证了理论结果。
A high order stabilization-free virtual element method for general second-order elliptic eigenvalue problem
本文提出一种高阶无稳定化虚拟元方法，用于解决一般二阶椭圆特征值问题，推导了近似特征空间和特征值的最优先验误差估计，并通过数值实验验证了方法的有效性。
A Virtual Element Method for the Transmission Eigenvalue Problem
本文提出一种虚拟元方法求解非自伴四阶本征值问题，通过C¹连续离散化并利用紧凑非自伴算子的经典近似理论，获得最优阶误差估计及本征值双阶精度，通过数值实验验证了方法的有效性。
The aim of this paper is to analyze the influence of small edges in the computation of the spectrum of the Steklov eigenvalue problem by a lowest order virtual element method. Under weaker assumptions on the polygonal meshes, which can permit arbitrarily small edges with respect to the element diameter, we show that the scheme provides a correct approximation of the spectrum and prove optimal error estimates for the eigenfunctions and a double order for the eigenvalues. Finally, we report some numerical tests supporting the theoretical results.
A noncoforming virtual element approximation for the Oseen eigenvalue problem
本文提出一种非协调虚拟元方法逼近二维Oseen特征值问题，基于紧算子理论证明收敛性与误差估计，并通过数值实验验证了其双重谱收敛阶。
In this paper we analyze a virtual element method (VEM) for a pseudostress formulation of the Stokes eigenvalue problem. This formulation allows to eliminate the velocity and the pressure, leading to an elliptic formulation where the only unknown is the pseudostress tensor. The velocity and pressure can be recovered by a post-process. Adapting the non-compact operator theory, we prove that our method provides a correct approximation of the spectrum and is spurious free. We prove a priori error estimates, with optimal order, which we confirm with some numerical tests.
The nonconforming virtual element method for eigenvalue problems
本文研究非符合虚拟元方法在椭圆特征值问题中的应用，提出两种离散问题形式并分析其收敛性，证明了特征函数和特征值的最优误差估计。
Virtual element approximation of eigenvalue problems: is the stabilization of the right hand side necessary?
本文证明在椭圆自共轭本征值问题中，采用低阶标准虚元空间时，质量矩阵的稳定化并非必要，数值结果表明更高阶方案在不同网格序列上也成立。
A posteriori error estimates for a Virtual Elements Method for the Steklov eigenvalue problem
本文研究了虚拟元方法在Steklov特征值问题中的后验误差分析，提出了一种残差型误差估计器，并通过数值实验验证了其有效性。
A virtual element approximation for the modified transmission eigenvalues for natural materials
针对逆散射中天然材料的修正透射特征值问题，利用T-强制性质建立离散源问题的适定性，给出先验误差估计，并通过数值实验验证方法的有效性。
Avoiding stabilization terms in virtual elements for eigenvalue problems: The Reduced Basis Virtual Element Method
本文提出缩减基虚拟元素法，用于求解特征值问题，通过避免稳定化项实现高精度谱近似。
A locking-free mixed virtual element discretization for the two dimensional elasticity eigenvalue problem
本文提出并分析了一种混合虚拟元方法，用于二维弹性本征值问题的本征值和本征函数近似，证明了离散解算子在网格尺寸趋于零时收敛到连续对应项，并通过数值实验验证了方法的无锁性和准确逼近谱的能力。
A posteriori analysis of a virtual element approach on polytopal meshes for the buckling eigenvalue problem
本文提出一种基于残差的后验误差估计器，用于解决板壳问题的 conforming C^1 虚拟元方法，考虑了二维和三维非线性平面应力效应，展示了其在多面体网格上的计算能力和在复杂板结构预测分析中的潜力。
The virtual element method for eigenvalue problems with potential terms on polytopal meshes
本文扩展了符合虚拟元方法，用于求解具有势项的特征值问题，重点探讨了该方法在量子谐振子问题和奇异特征值问题中的应用，展示了方法的灵活性和最优收敛性。
The $p$- and $hp$-versions of the virtual element method for elliptic eigenvalue problems
本文研究了虚拟元方法的$p$-和$hp$-版本，用于近似具有势项的椭圆算子的特征对，通过Schrödinger方程的伪势项应用展示了该模型，并详细分析了$p$-版本方法的理论结果，证明了在解析特征函数情况下具有指数收敛性，同时通过大量实验验证了在特征函数具有有限Sobolev正则性时，通过$hp$-细化可获得特征值关于自由度数立方根的指数近似。
Conforming and non-conforming virtual element methods for the biharmonic Steklov eigenvalue problem with minimum regularity
针对一般形状多边形域上的四阶Steklov特征值问题，提出协调和C^0非协调虚拟元方法，通过富化算子进行收敛分析，证明无伪特征值并得到最优收敛阶。
We present a new method to construct Virtual Element spaces on polygons with curved edges.
We present two kinds of lowest-order virtual element methods for planar linear elasticity problems. For the first one we use the nonconforming virtual element method with a stabilizing term. It can be interpreted as a modification of the nonconforming Crouzeix-Raviart finite element method as suggested in [22] to the virtual element method. For the second one we use the conforming virtual element for one component of the displacement vector and the nonconforming virtual element for the other. This approach can be seen as an extension of the idea of Kouhia and Stenberg suggested in [23] to the …
The maximum norm error estimations for virtual element methods are studied. To establish the error estimations, we prove higher local regularity based on delicate analysis of Green's functions and high-order local error estimations for the partition of the virtual element solutions. The maximum norm of the exact gradient and the gradient of the projection of the virtual element solutions are proved to achieve optimal convergence results. For high-order virtual element methods, we establish the optimal convergence results in $L^{\infty}$ norm. Our theoretical discoveries are validated by a nume…
Some Error Analysis on Virtual Element Methods
本文针对多边形网格上的虚拟元方法，研究反向不等式、范数等价性和插值误差估计。
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 consider the Virtual Element discretization of the Navier-Stokes equations coupled with the heat equation where the viscosity depends on the temperature. We present the Virtual Element discretization of the coupled problem, show its well-posedness, and prove optimal error estimates. Numerical experiments which confirm the theoretical error bounds are also presented.
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.
The Neural Approximated Virtual Element Method for Elasticity Problems
针对弹性问题的数值求解，提出结合有限元法、虚拟元法与深度神经网络的神经近似虚拟元方法，通过神经网络逐元素近似虚拟基函数简化离散化，经线性和非线性弹性问题测试验证其处理非线性的优势。
In this paper, we propose a new finite element approach, which is different than the classic Babuska-Osborn theory, to approximate Dirichlet eigenvalues. The Dirichlet eigenvalue problem is formulated as the eigenvalue problem of a holomorphic Fredholm operator function of index zero. Using conforming finite elements, the convergence is proved using the abstract approximation theory for holomorphic operator functions. The spectral indicator method is employed to compute the eigenvalues. A numerical example is presented to validate the theory.
We present a simple and efficient MATLAB implementation of the linear virtual element method for the three dimensional Poisson equation. The purpose of this software is primarily educational, to demonstrate how the key components of the method can be translated into code.
Reduced basis stabilization and post-processing for the virtual element method
提出约化基方法构造虚拟元空间节点基函数的近似，用于稳定项设计与解的后处理。
Non-conforming finite element methods for transmission eigenvalue problem
本文研究了传输本征值问题的非符合有限元方法，推导了非自伴随线性本征值问题的弱形式，并证明了Adini元、Morley-Zienkiewicz元等的误差估计，通过数值实验验证了方法的有效性。
Asymptotic Expansions and Extrapolation of Approximate Eigenvalues for Second Order Elliptic Problems by Mixed Finite Element Methods
针对一般二阶椭圆特征值问题，采用最低阶Raviart-Thomas混合有限元方法，推导了特征值近似的渐近误差展开，并基于此进行外推以提高精度，同时证明了混合有限元方法中特征值问题的有限元投影与有限元近似之间的超收敛性质。
We consider finite element approximations of the Maxwell eigenvalue problem in two dimensions. We prove, in certain settings, convergence of the discrete eigenvalues using Lagrange finite elements. In particular, we prove convergence in three scenarios: piecewise linear elements on Powell--Sabin triangulations, piecewise quadratic elements on Clough--Tocher triangulations, and piecewise quartics (and higher) elements on general shape-regular triangulations. We provide numerical experiments that support the theoretical results. The computations also show that, on general triangulations, the eig…
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.
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…
We introduce a nonconforming virtual element method for the Poisson equation on domains with curved boundary and internal interfaces. We prove arbitrary order optimal convergence in the energy and $L^2$ norms, and validate the theoretical results with numerical experiments. Compared to existing nodal virtual elements on curved domains, the proposed scheme has the advantage that it can be designed in any dimension.
The lowest-order Neural Approximated Virtual Element Method on polygonal elements
本文提出最低阶神经近似虚拟元方法，用神经网络局部近似虚拟元基函数以消除稳定化和投影算子问题，并通过数值实验验证其在一般多边形网格及带悬挂节点三角形网格上的优势。
We present the construction and application of a first order stabilization-free virtual element method to problems in plane elasticity. Well-posedness and error estimates of the discrete problem are established. The method is assessed on a series of well-known benchmark problems from linear elasticity and numerical results are presented that affirm the optimal convergence rate of the virtual element method in the $L^2$ norm and the energy seminorm.
The lower bound property of the Morley element eigenvalues
本文证明了在正则网格上，包括自适应局部细化网格，莫里元素特征值从下界逼近第四阶椭圆特征值问题的精确解，针对夹紧边界条件在任意维度的问题，并实现了自适应计算以获得夹紧板在张力下的振动问题的莫里元素特征值下界。
Convergence of adaptive finite element methods for eigenvalue problems
本文证明了对二阶椭圆特征值问题，自适应有限元方法在最小标记单元细化、合理标记策略和任意初始三角剖分下，对于简单和多重特征值均收敛。
Regular convergence, together with various other types of convergence, has been studied since the 1970s for the discrete approximations of linear operators. In this paper, we consider the eigenvalue approximation of compact operators whose spectral problem can be written as the eigenvalue problem of some holomophic Fredholm operator function. Focusing on the finite element methods (conforming, discontinuous Galerkin, etc.), we show that the regular convergence of discrete holomorphic operator functions follows from the approximation property of the finite element spaces and the compact converg…
The lowest-order Neural Approximated Virtual Element Method
本文提出最低阶神经近似虚拟元方法，利用神经网络替代虚拟元法中的投影与稳定化算子，并通过椭圆型问题数值实验验证其有效性。
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.
Virtual element methods for a quad-curl problem on general planar domains
本文设计并分析了基于无旋场霍奇分解的虚拟元方法，用于解决一般多边形域中的四旋问题，并展示了支持理论分析的数值结果。
In this paper, we introduce a new Virtual Element Method (VEM) not requiring any stabilization term based on the usual enhanced first-order VEM space. The new method relies on a modified formulation of the discrete diffusion operator that ensures stability preserving all the properties of the differential operator.
The modified Maxwell's Stekloff eigenvalue problem arises recently from the inverse electromagnetic scattering theory for inhomogeneous media. This paper contains a rigorous analysis of both the eigenvalue problem and the associated source problem on Lipschitz polyhedra. A new finite element method is proposed to compute Stekloff eigenvalues. By applying the Babuska-Osborn theory, we prove an error estimate without additional regularity assumptions. Numerical results are presented for validation.
The weak Galerkin finite element method for the Steklov eigenvalue problem
本文将弱 Galerkin 有限元方法用于 Steklov 特征值问题，借助非协调有限元空间获得任意高阶且有保证的特征值下界，并通过数值实验验证其准确性。
When rational functions meet virtual elements: The lightning Virtual Element Method
本文提出闪电虚拟元方法，用极点指数聚集于单元角点附近的有理函数近似VEM基函数的虚拟分量以消除稳定化项，从而简化误差分析并可直接获取逐点数值解，且具有良好的并行性和成本效率。
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…
Equivalence of Weak Galerkin Methods and Virtual Element Methods for Elliptic Equations
本文提出弱伽辽金方法的改进版本，并证明其与虚拟元方法的新版本等价，同时展示原弱伽辽金方法与非符合虚拟元方法的等价性，通过自由度的度量实现方法间的转换。
We build and analyze Balancing Domain Decomposition by Constraint (BDDC) and Finite Element Tearing and Interconnecting Dual Primal (FETI-DP) preconditioners for elliptic problems discretized by the virtual element method (VEM). We prove polylogarithmic condition number bounds, independent of the number of subdomains, the mesh size, and jumps in the diffusion coefficients. Numerical experiments confirm the theory.
In this work we analyze a virtual element method on polyhedral meshes for solving the sixth-order elliptic problem with simply supported boundary conditions. We apply the Ciarlet-Raviart arguments to introduce an auxiliary unknown $σ:=-Δ^2 u$ and to search the main uknown $u$ in the $H^2\cap H_0^1$ Sobolev space. The virtual element discretization is well possed on a $C^1\times C^0$ virtual element spaces. We also provide the convergence and error estimates results. Finally, we report a series of numerical tests to verify the performance of numerical scheme.
Virtual Element Methods Without Extrinsic Stabilization
本文提出任意阶无需外部稳定化的虚拟元方法，通过构造局部H(div)协调宏观有限元空间保证梯度投影可计算且具有一致下界，并给出最优误差估计与数值验证。
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…
A Multilevel Correction Type of Adaptive Finite Element Method for Eigenvalue Problems
提出一种基于多层校正方案的自适应有限元方法，通过求解自适应网格上的边值问题和粗网格上的小规模特征值问题，提高特征值问题求解效率，并理论验证收敛性与最优复杂度。
We present a Virtual Element Method for the 3D linear elasticity problems, based on Hellinger-Reissner variational principle. In the framework of the small strain theory, we propose a low-order scheme with a-priori symmetric stresses and continuous tractions across element interfaces. A convergence and stability analysis is developed and we confirm the theoretical predictions via some numerical tests.
The paper presents a finite element scheme for the elastic transmission eigenvalue problem written as a fourth order eigenvalue problem. The scheme uses piecewise cubic polynomials and obtains optimal convergence rate. Compared with other low-degree and nonconforming finite element schemes, the scheme inherits the continuous bilinear form which does not need extra stabilizations and is thus simple to implement.
The Reduced Basis Multigrid scheme for the Virtual Element Method
本文提出了一种基于降阶基方法的非嵌套W-cycle多网格方案，用于在多边形网格上实现最低阶虚拟元方法，以解决虚拟元空间隐式定义带来的计算问题。
Adaptive finite element method for the Maxwell eigenvalue problem
本文证明了基于边有限元的标准自适应方案在近似Maxwell方程特征值问题解时的最优收敛性，通过将问题等价转换为混合特征值问题进行证明。
In this paper we consider the Virtual Element discretization of a minimal surface problem, a quasi-linear elliptic partial differential equation modeling the problem of minimizing the area of a surface subject to a prescribed boundary condition. We derive optimal error estimate and present several numerical tests assessing the validity of the theoretical results.
In this letter we compare the behaviour of standard Virtual Element Methods (VEM) and stabilization free Enlarged Enhancement Virtual Element Methods (E$^2$VEM) with the focus on some elliptic test problems whose solution and diffusivity tensor are characterized by anisotropies. Results show that the possibility to avoid an arbitrary stabilizing part, offered by E$^2$VEM methods, can reduce the magnitude of the error on general polygonal meshes and help convergence.
Nodal auxiliary space preconditioners for mixed virtual element methods
本文提出针对最低阶面片和边虚拟元的节点辅助空间预条件子，通过离散正则分解推广Hiptmair-Xu方法，证明谱条件数有界且不依赖网格尺寸，并在数值实验中验证了其对高纵横比网格的鲁棒性。
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