scholarly journals The harmonic virtual element method: stabilization and exponential convergence for the Laplace problem on polygonal domains

2018 ◽  
Vol 39 (4) ◽  
pp. 1787-1817 ◽  
Author(s):  
Alexey Chernov ◽  
Lorenzo Mascotto
Keyword(s):  
Finite Element ◽  
Laplace Equation ◽  
Degrees Of Freedom ◽  
Exponential Convergence ◽  
Harmonic Polynomial ◽  
Harmonic Polynomials ◽  
Virtual Element Method ◽  
Polygonal Domains ◽  
Virtual Element ◽  
Element Method

Abstract We introduce the harmonic virtual element method (VEM) (harmonic VEM), a modification of the VEM (Beirão da Veiga et al. (2013) Basic principles of virtual element methods. Math. Models Methods Appl. Sci., 23, 199–214.) for the approximation of the two-dimensional Laplace equation using polygonal meshes. The main difference between the harmonic VEM and the VEM is that in the former method only boundary degrees of freedom are employed. Such degrees of freedom suffice for the construction of a proper energy projector on (piecewise harmonic) polynomial spaces. The harmonic VEM can also be regarded as an ‘$H^1$-conformisation’ of the Trefftz discontinuous Galerkin-finite element method (TDG-FEM) (Hiptmair et al. (2014) Approximation by harmonic polynomials in starshaped domains and exponential convergence of Trefftz hp-DGFEM. ESAIM Math. Model. Numer. Anal., 48, 727–752.). We address the stabilization of the proposed method and develop an hp version of harmonic VEM for the Laplace equation on polygonal domains. As in TDG-FEM, the asymptotic convergence rate of harmonic VEM is exponential and reaches order $\mathscr{O}(\exp (-b\sqrt [2]{N}))$, where $N$ is the number of degrees of freedom. This result overperforms its counterparts in the framework of hp FEM (Schwab, C. (1998)p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Clarendon Press Oxford.) and hp VEM (Beirão da Veiga et al. (2018) Exponential convergence of the hp virtual element method with corner singularity. Numer. Math., 138, 581–613.), where the asymptotic rate of convergence is of order $\mathscr{O}(\exp(-b\sqrt [3]{N}))$.

BASIC PRINCIPLES OF VIRTUAL ELEMENT METHODS

2012 ◽  
Vol 23 (01) ◽  
pp. 199-214 ◽  
Author(s):  
L. BEIRÃO DA VEIGA ◽  
F. BREZZI ◽  
A. CANGIANI ◽  
G. MANZINI ◽  
L. D. MARINI ◽  
...  
Keyword(s):  
Finite Element ◽  
Degrees Of Freedom ◽  
Academic Press ◽  
Polynomial Functions ◽  
Generalized Finite Element Method ◽  
Virtual Element Method ◽  
Order Continuity ◽  
Virtual Element ◽  
Basic Features ◽  
Element Method

We present, on the simplest possible case, what we consider as the very basic features of the (brand new) virtual element method. As the readers will easily recognize, the virtual element method could easily be regarded as the ultimate evolution of the mimetic finite differences approach. However, in their last step they became so close to the traditional finite elements that we decided to use a different perspective and a different name. Now the virtual element spaces are just like the usual finite element spaces with the addition of suitable non-polynomial functions. This is far from being a new idea. See for instance the very early approach of E. Wachspress [A Rational Finite Element Basic (Academic Press, 1975)] or the more recent overview of T.-P. Fries and T. Belytschko [The extended/generalized finite element method: An overview of the method and its applications, Int. J. Numer. Methods Engrg.84 (2010) 253–304]. The novelty here is to take the spaces and the degrees of freedom in such a way that the elementary stiffness matrix can be computed without actually computing these non-polynomial functions, but just using the degrees of freedom. In doing that we can easily deal with complicated element geometries and/or higher-order continuity conditions (like C1, C2, etc.). The idea is quite general, and could be applied to a number of different situations and problems. Here however we want to be as clear as possible, and to present the simplest possible case that still gives the flavor of the whole idea.

scholarly journals Exponential convergence of the hp virtual element method in presence of corner singularities

2017 ◽  
Vol 138 (3) ◽  
pp. 581-613 ◽  
Author(s):  
L. Beirão da Veiga ◽  
A. Chernov ◽  
L. Mascotto ◽  
A. Russo
Keyword(s):  
Exponential Convergence ◽  
Corner Singularities ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals Bulk-surface virtual element method for systems of PDEs in two-space dimensions

2021 ◽  
Vol 147 (2) ◽  
pp. 305-348
Author(s):  
Massimo Frittelli ◽  
Anotida Madzvamuse ◽  
Ivonne Sgura
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Boundary Conditions ◽  
Virtual Element Method ◽  
Link Type ◽  
Bulk Surface ◽  
Two Space Dimensions ◽  
Virtual Element ◽  
Special Case ◽  
Element Method

AbstractIn this paper we consider a coupled bulk-surface PDE in two space dimensions. The model consists of a PDE in the bulk that is coupled to another PDE on the surface through general nonlinear boundary conditions. For such a system we propose a novel method, based on coupling a virtual element method (Beirão Da Veiga et al. in Math Models Methods Appl Sci 23(01):199–214, 2013. https://doi.org/10.1051/m2an/2013138) in the bulk domain to a surface finite element method (Dziuk and Elliott in Acta Numer 22:289–396, 2013. https://doi.org/10.1017/s0962492913000056) on the surface. The proposed method, which we coin the bulk-surface virtual element method includes, as a special case, the bulk-surface finite element method (BSFEM) on triangular meshes (Madzvamuse and Chung in Finite Elem Anal Des 108:9–21, 2016. https://doi.org/10.1016/j.finel.2015.09.002). The method exhibits second-order convergence in space, provided the exact solution is $$H^{2+1/4}$$ H 2 + 1 / 4 in the bulk and $$H^2$$ H 2 on the surface, where the additional $$\frac{1}{4}$$ 1 4 is required only in the simultaneous presence of surface curvature and non-triangular elements. Two novel techniques introduced in our analysis are (i) an $$L^2$$ L 2 -preserving inverse trace operator for the analysis of boundary conditions and (ii) the Sobolev extension as a replacement of the lifting operator (Elliott and Ranner in IMA J Num Anal 33(2):377–402, 2013. https://doi.org/10.1093/imanum/drs022) for sufficiently smooth exact solutions. The generality of the polygonal mesh can be exploited to optimize the computational time of matrix assembly. The method takes an optimised matrix-vector form that also simplifies the known special case of BSFEM on triangular meshes (Madzvamuse and Chung 2016). Three numerical examples illustrate our findings.

scholarly journals Vertex Displacement-Based Discontinuous Deformation Analysis Using Virtual Element Method

Materials ◽  
2021 ◽  
Vol 14 (5) ◽  
pp. 1252
Author(s):  
Hongming Luo ◽  
Guanhua Sun ◽  
Lipeng Liu ◽  
Wei Jiang
Keyword(s):  
Degrees Of Freedom ◽  
Discontinuous Deformation Analysis ◽  
Deformation Analysis ◽  
Virtual Element Method ◽  
Time Step ◽  
Element Space ◽  
Displacement Mode ◽  
Discontinuous Deformation ◽  
Virtual Element ◽  
Element Method

To avoid disadvantages caused by rotational degrees of freedom in the original Discontinuous Deformation Analysis (DDA), a new block displacement mode is defined within a time step, where displacements of all the block vertices are taken as the degrees of freedom. An individual virtual element space V1(Ω) is defined for a block to illustrate displacement of the block using the Virtual Element Method (VEM). Based on VEM theory, the total potential energy of the block system in DDA is formulated and minimized to obtain the global equilibrium equations. At the end of a time step, the vertex coordinates are updated by adding their incremental displacement to their previous coordinates. In the new method, no explicit expression for the displacement u is required, and all numerical integrations can be easily computed. Four numerical examples originally designed by Shi are analyzed, verifying the effectiveness and precision of the proposed method.

scholarly journals A virtual element generalization on polygonal meshes of the Scott-Vogelius finite element method for the 2-D Stokes problem

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Gianmarco Manzini ◽  
Annamaria Mazzia
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Approximation ◽  
Convergence Rates ◽  
Stokes Problem ◽  
Element Approximation ◽  
Virtual Element Method ◽  
Polygonal Meshes ◽  
Virtual Element ◽  
Element Method

<p style='text-indent:20px;'>The Virtual Element Method (VEM) is a Galerkin approximation method that extends the Finite Element Method (FEM) to polytopal meshes. In this paper, we present a conforming formulation that generalizes the Scott-Vogelius finite element method for the numerical approximation of the Stokes problem to polygonal meshes in the framework of the virtual element method. In particular, we consider a straightforward application of the virtual element approximation space for scalar elliptic problems to the vector case and approximate the pressure variable through discontinuous polynomials. We assess the effectiveness of the numerical approximation by investigating the convergence on a manufactured solution problem and a set of representative polygonal meshes. We numerically show that this formulation is convergent with optimal convergence rates except for the lowest-order case on triangular meshes, where the method coincides with the <inline-formula><tex-math id="M1">\begin{document}$ {\mathbb{P}}_{{1}}-{\mathbb{P}}_{{0}} $\end{document}</tex-math></inline-formula> Scott-Vogelius scheme, and on square meshes, which are situations that are well-known to be unstable.</p>

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