scholarly journals Bridging the hybrid high-order and virtual element methods

2020 ◽  
Author(s):  
Simon Lemaire
Keyword(s):  
High Order ◽  
Virtual Space ◽  
Approximation Properties ◽  
Virtual Spaces ◽  
Local Polynomial ◽  
Poisson Problem ◽  
Virtual Element Methods ◽  
Dimension 2 ◽  
Virtual Element ◽  
Polytopal Meshes

Abstract We present a unifying viewpoint on hybrid high-order and virtual element methods on general polytopal meshes in dimension $2$ or $3$, in terms of both formulation and analysis. We focus on a model Poisson problem. To build our bridge (i) we transcribe the (conforming) virtual element method into the hybrid high-order framework and (ii) we prove $H^m$ approximation properties for the local polynomial projector in terms of which the local virtual element discrete bilinear form is defined. This allows us to perform a unified analysis of virtual element/hybrid high-order methods, that differs from standard virtual element analyses by the fact that the approximation properties of the underlying virtual space are not explicitly used. As a complement to our unified analysis we also study interpolation in local virtual spaces, shedding light on the differences between the conforming and nonconforming cases.

Some Estimates for Virtual Element Methods

2017 ◽  
Vol 17 (4) ◽  
pp. 553-574 ◽  
Author(s):  
Susanne C. Brenner ◽  
Qingguang Guan ◽  
Li-Yeng Sung
Keyword(s):  
Error Estimates ◽  
Three Dimensions ◽  
Poisson Problem ◽  
Polyhedral Meshes ◽  
Virtual Element Methods ◽  
Virtual Element

AbstractWe present novel techniques for obtaining the basic estimates of virtual element methods in terms of the shape regularity of polygonal/polyhedral meshes. We also derive new error estimates for the Poisson problem in two and three dimensions.

scholarly journals Virtual element methods on meshes with small edges or faces

2018 ◽  
Vol 28 (07) ◽  
pp. 1291-1336 ◽  
Author(s):  
Susanne C. Brenner ◽  
Li-Yeng Sung
Keyword(s):  
Stability Analysis ◽  
Three Dimensions ◽  
Two Dimensional ◽  
Virtual Element Method ◽  
Basic Principles ◽  
Poisson Problem ◽  
Polyhedral Meshes ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
Element Method

We consider a model Poisson problem in [Formula: see text] ([Formula: see text]) and establish error estimates for virtual element methods on polygonal or polyhedral meshes that can contain small edges ([Formula: see text]) or small faces ([Formula: see text]). Our results extend the ones in [L. Beirão da Veiga, C. Lovadina and A. Russo, Stability analysis for the virtual element method, Math. Models Methods Appl. Sci. 27 (2017) 2557–2594] for the original two-dimensional virtual element method from [L. Beirão da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini and A. Russo, Basic principles of virtual element methods, Math. Models Methods Appl. Sci. 23 (2013) 199–214] to the version of the virtual element method in [B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini and A. Russo, Equivalent projectors for virtual element methods, Comput. Math. Appl. 66 (2013) 376–391] that can also be applied to problems in three dimensions.

scholarly journals Robust Hybrid High-Order Method on Polytopal Meshes with Small Faces

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jérôme Droniou ◽  
Liam Yemm
Keyword(s):  
Error Estimates ◽  
Relative Size ◽  
High Order ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Poisson Problem ◽  
Large Numbers ◽  
Number Of Faces ◽  
2D And 3D ◽  
Polytopal Meshes

Abstract We design a Hybrid High-Order (HHO) scheme for the Poisson problem that is fully robust on polytopal meshes in the presence of small edges/faces. We state general assumptions on the stabilisation terms involved in the scheme, under which optimal error estimates (in discrete and continuous energy norms, as well as L 2 L^{2} -norm) are established with multiplicative constants that do not depend on the maximum number of faces in each element, or the relative size between an element and its faces. We illustrate the error estimates through numerical simulations in 2D and 3D on meshes designed by agglomeration techniques (such meshes naturally have elements with a very large numbers of faces, and very small faces).

scholarly journals Ws,p-approximation properties of elliptic projectors on polynomial spaces, with application to the error analysis of a Hybrid High-Order discretisation of Leray–Lions problems

2017 ◽  
Vol 27 (05) ◽  
pp. 879-908 ◽  
Author(s):  
Daniele A. Di Pietro ◽  
Jérôme Droniou
Keyword(s):  
Turbulent Flows ◽  
Approximation Error ◽  
High Order ◽  
Approximation Properties ◽  
Weak Formulation ◽  
General Applicability ◽  
Local Polynomial ◽  
Discrete Norm ◽  
Orthogonal Projectors ◽  
Glacier Motion

In this work, we prove optimal [Formula: see text]-approximation estimates (with [Formula: see text]) for elliptic projectors on local polynomial spaces. The proof hinges on the classical Dupont–Scott approximation theory together with two novel abstract lemmas: An approximation result for bounded projectors, and an [Formula: see text]-boundedness result for [Formula: see text]-orthogonal projectors on polynomial subspaces. The [Formula: see text]-approximation results have general applicability to (standard or polytopal) numerical methods based on local polynomial spaces. As an illustration, we use these [Formula: see text]-estimates to derive novel error estimates for a Hybrid High-Order (HHO) discretisation of Leray–Lions elliptic problems whose weak formulation is classically set in [Formula: see text] for some [Formula: see text]. This kind of problems appears, e.g. in the modelling of glacier motion, of incompressible turbulent flows, and in airfoil design. Denoting by [Formula: see text] the meshsize, we prove that the approximation error measured in a [Formula: see text]-like discrete norm scales as [Formula: see text] when [Formula: see text] and as [Formula: see text] when [Formula: see text].

scholarly journals Robust nonconforming virtual element methods for general fourth-order problems with varying coefficients

2021 ◽  
Author(s):  
Andreas Dedner ◽  
Alice Hodson
Keyword(s):  
Fourth Order ◽  
Perturbation Parameter ◽  
Two Dimensions ◽  
Optimal Error Estimates ◽  
Projection Operators ◽  
Varying Coefficients ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
Perturbation Problems ◽  
Fourth Order Problems

Abstract We present a class of nonconforming virtual element methods for general fourth-order partial differential equations in two dimensions. We develop a generic approach for constructing the necessary projection operators and virtual element spaces. Optimal error estimates in the energy norm are provided for general linear fourth-order problems with varying coefficients. We also discuss fourth-order perturbation problems and present a novel nonconforming scheme which is uniformly convergent with respect to the perturbation parameter without requiring an enlargement of the space. Numerical tests are carried out to verify the theoretical results. We conclude with a brief discussion on how our approach can easily be applied to nonlinear fourth-order problems.

scholarly journals A Hybrid High-Order method for Kirchhoff–Love plate bending problems

2018 ◽  
Vol 52 (2) ◽  
pp. 393-421 ◽  
Author(s):  
Francesco Bonaldi ◽  
Daniele A. Di Pietro ◽  
Giuseppe Geymonat ◽  
Françoise Krasucki
Keyword(s):  
Sharp Estimate ◽  
High Order ◽  
Plate Bending ◽  
Mechanical Modeling ◽  
High Order Method ◽  
Local Polynomial ◽  
Bending Behavior ◽  
Bending Problems ◽  
Norm Of The Error ◽  
Theoretical Results

We present a novel Hybrid High-Order (HHO) discretization of fourth-order elliptic problems arising from the mechanical modeling of the bending behavior of Kirchhoff–Love plates, including the biharmonic equation as a particular case. The proposed HHO method supports arbitrary approximation orders on general polygonal meshes, and reproduces the key mechanical equilibrium relations locally inside each element. When polynomials of degree k ≥ 1 are used as unknowns, we prove convergence in hk+1 (with h denoting, as usual, the meshsize) in an energy-like norm. A key ingredient in the proof are novel approximation results for the energy projector on local polynomial spaces. Under biharmonic regularity assumptions, a sharp estimate in hk+3 is also derived for the L2-norm of the error on the deflection. The theoretical results are supported by numerical experiments, which additionally show the robustness of the method with respect to the choice of the stabilization.

scholarly journals Embates discursivos e significantes vazios nas manifestações de junho de 2013 / Discursive struggles and empty signifiers at the demonstrations of June 2013

2017 ◽  
Vol 7 (1) ◽  
pp. 101
Author(s):  
Henrique De Oliveira Lee ◽  
Camila Rodrigues Francisco
Keyword(s):  
Virtual Space ◽  
Political Identities ◽  
Virtual Spaces ◽  
Ernesto Laclau ◽  
Chantal Mouffe ◽  
The Many

ResumoO objetivo deste trabalho é propor uma investigação sobre os embates discursivos entre diversas identidades políticas, encenados nas ruas e no espaço virtual, durante as manifestações realizadas nas capitais brasileiras durante o mês de junho de 2013. Através do pensamento de Ernesto Laclau (1994) e Chantal Mouffe (1990), especificamente em torno dos conceitos de “antagonismo” e “significantes vazios”, propõe-se uma análise de determinadas produções e recepções de enunciados do embate estabelecido entre os diversos grupos que participaram das manifestações realizadas nas capitais brasileiras durante o mês de junho de 2013. Os significantes vazios  permitiram verificar a emergência de significados antagônicos nas produções discursivas surgidas nas ruas e nos ambientes virtuais como um modo de articulação de antagonismos e identidades políticas.Palavras-chave: Laclau; Significante vazio; Antagonismo. AbstractThe goal of this paper is to propose an inquiry about the discursive struggle performed at the streets and at the virtual space among the many political identities present in the demonstrations of June 2013 in Brazil´s biggest cities. It is proposed an analysis of certain production and reception of utterances of the struggle established among the many groups that took part in the demonstration through the framework of Laclau´s and Mouffe´s concepts of “antagonism” and “empty signifiers”. The latter allowed us to verify the emergency of antagonistic meanings in the discursive production found in the streets and virtual spaces as a way to articulate antagonism and political identities.      Keywords: Laclau; Empty signifiers; Antagonism.

scholarly journals A nonconforming Trefftz virtual element method for the Helmholtz problem

2019 ◽  
Vol 29 (09) ◽  
pp. 1619-1656 ◽  
Author(s):  
Lorenzo Mascotto ◽  
Ilaria Perugia ◽  
Alexander Pichler
Keyword(s):  
Convergence Rates ◽  
Plane Waves ◽  
Local Approximation ◽  
Approximation Properties ◽  
Virtual Element Method ◽  
Impedance Boundary ◽  
Helmholtz Problem ◽  
Text Version ◽  
Virtual Element ◽  
Element Method

We introduce a novel virtual element method (VEM) for the two-dimensional Helmholtz problem endowed with impedance boundary conditions. Local approximation spaces consist of Trefftz functions, i.e. functions belonging to the kernel of the Helmholtz operator. The global trial and test spaces are not fully discontinuous, but rather interelement continuity is imposed in a nonconforming fashion. Although their functions are only implicitly defined, as typical of the VEM framework, they contain discontinuous subspaces made of functions known in closed form and with good approximation properties (plane-waves, in our case). We carry out an abstract error analysis of the method, and derive [Formula: see text]-version error estimates. Moreover, we initiate its numerical investigation by presenting a first test, which demonstrates the theoretical convergence rates.

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