The curl–curl conforming virtual element method for the quad-curl problem

2021 ◽  
pp. 1-32
Author(s):  
Jikun Zhao ◽  
Bei Zhang
Keyword(s):  
Two Dimensions ◽  
Virtual Element Method ◽  
Rham Complex ◽  
Order Case ◽  
De Rham ◽  
Virtual Element ◽  
The Stability ◽  
Discrete Formulation ◽  
Theoretical Results ◽  
Discrete Complex

In this paper, we present the [Formula: see text]-conforming virtual element (VE) method for the quad-curl problem in two dimensions. Based on the idea of de Rham complex, we first construct three families of [Formula: see text]-conforming VEs, of which the simplest one has only one degree of freedom associated to each vertex and each edge in the lowest-order case, respectively. An exact discrete complex is established between the [Formula: see text]-conforming and [Formula: see text]-conforming VEs. We rigorously prove the interpolation error estimates, the stability of discrete bilinear forms, the coercivity and inf–sup condition of the corresponding discrete formulation. We show that the conforming VEs have the optimal convergence. Some numerical examples are given to confirm the theoretical results.

scholarly journals A Virtual Element Method for the Wave Equation on Curved Edges in Two Dimensions

2021 ◽  
Vol 90 (1) ◽  
Author(s):  
Franco Dassi ◽  
Alessio Fumagalli ◽  
Ilario Mazzieri ◽  
Anna Scotti ◽  
Giuseppe Vacca
Keyword(s):  
Wave Equation ◽  
Two Dimensions ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals The Virtual Element Method with curved edges

2019 ◽  
Vol 53 (2) ◽  
pp. 375-404 ◽  
Author(s):  
L. Beirão da Veiga ◽  
A. Russo ◽  
G. Vacca
Keyword(s):  
Rate Of Convergence ◽  
Two Dimensions ◽  
Optimal Rate ◽  
Virtual Element Method ◽  
Curved Boundary ◽  
Optimal Rate Of Convergence ◽  
Virtual Element ◽  
Curved Interface ◽  
Element Method

In this paper we initiate the investigation of Virtual Elements with curved faces. We consider the case of a fixed curved boundary in two dimensions, as it happens in the approximation of problems posed on a curved domain or with a curved interface. While an approximation of the domain with polygons leads, for degree of accuracy k≥2, to a sub-optimal rate of convergence, we show (both theoretically and numerically) that the proposed curved VEM lead to an optimal rate of convergence.

scholarly journals The mixed virtual element method on curved edges in two dimensions

2021 ◽  
Vol 386 ◽  
pp. 114098
Author(s):  
Franco Dassi ◽  
Alessio Fumagalli ◽  
Davide Losapio ◽  
Stefano Scialò ◽  
Anna Scotti ◽  
...  
Keyword(s):  
Two Dimensions ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals Stability analysis for the virtual element method

2017 ◽  
Vol 27 (13) ◽  
pp. 2557-2594 ◽  
Author(s):  
Lourenço Beirão da Veiga ◽  
Carlo Lovadina ◽  
Alessandro Russo
Keyword(s):  
Degrees Of Freedom ◽  
Stability And Convergence ◽  
Virtual Element Method ◽  
Theoretical Predictions ◽  
Elliptic Model ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
The Stability ◽  
The One ◽  
General Meshes

We analyze the virtual element methods (VEM) on a simple elliptic model problem, allowing for more general meshes than the one typically considered in the VEM literature. For instance, meshes with arbitrarily small edges (with respect to the parent element diameter) can be dealt with. Our general approach applies to different choices of the stability form, including, for example, the “classical” one introduced in Ref. 4, and a recent one presented in Ref. 34. Finally, we show that the stabilization term can be simplified by dropping the contribution of the internal-to-the-element degrees of freedom. The resulting stabilization form, involving only the boundary degrees of freedom, can be used in the VEM scheme without affecting the stability and convergence properties. The numerical tests are in accordance with the theoretical predictions.

scholarly journals A polygonal discontinuous Galerkin method with minus one stabilization

2020 ◽  
Author(s):  
Silvia Bertoluzza ◽  
Daniele Prada
Keyword(s):  
Galerkin Method ◽  
Discontinuous Galerkin ◽  
Discontinuous Galerkin Method ◽  
Two Dimensions ◽  
Optimal Error Estimates ◽  
Virtual Element Method ◽  
Optimal Error ◽  
Numerical Tests ◽  
Virtual Element ◽  
Weak Shape

We propose a Discontinuous Galerkin method for the Poisson equation on polygonal tessellations in two dimensions, stabilized by penalizing, locally in each element K, a residual term involving the fluxes, measured in the norm of the dual of H 1 (K). The scalar product corresponding to such a norm is numerically realized via the introduction of a (minimal) auxiliary space  inspired by the Virtual Element Method. Stability and optimal error estimates in the broken H 1  norm are proven under a weak shape regularity assumption allowing the presence of very small edges. The results of numerical tests confirm the theoretical estimates.

scholarly journals Arbitrary-order intrinsic virtual element method for elliptic equations on surfaces

CALCOLO ◽  
2021 ◽  
Vol 58 (3) ◽  
Author(s):  
Elena Bachini ◽  
Gianmarco Manzini ◽  
Mario Putti
Keyword(s):  
Arbitrary Order ◽  
Surface Geometry ◽  
Virtual Element Method ◽  
Anisotropic Character ◽  
Local Reference System ◽  
Local Reference ◽  
Local Parametrization ◽  
Manufactured Solution ◽  
Virtual Element ◽  
Element Method

AbstractWe develop a geometrically intrinsic formulation of the arbitrary-order Virtual Element Method (VEM) on polygonal cells for the numerical solution of elliptic surface partial differential equations (PDEs). The PDE is first written in covariant form using an appropriate local reference system. The knowledge of the local parametrization allows us to consider the two-dimensional VEM scheme, without any explicit approximation of the surface geometry. The theoretical properties of the classical VEM are extended to our framework by taking into consideration the highly anisotropic character of the final discretization. These properties are extensively tested on triangular and polygonal meshes using a manufactured solution. The limitations of the scheme are verified as functions of the regularity of the surface and its approximation.

scholarly journals Dynamics of the rumor-spreading model with hesitation mechanism in heterogenous networks and bilingual environment

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Shuai Yang ◽  
Haijun Jiang ◽  
Cheng Hu ◽  
Juan Yu ◽  
Jiarong Li
Keyword(s):  
Lyapunov Method ◽  
Reproduction Number ◽  
Model Parameters ◽  
Rumor Spreading ◽  
Spreading Model ◽  
Reductio Ad Absurdum ◽  
The Stability ◽  
The Impact ◽  
Theoretical Results ◽  
The Basic Reproduction Number

Abstract In this paper, a novel rumor-spreading model is proposed under bilingual environment and heterogenous networks, which considers that exposures may be converted to spreaders or stiflers at a set rate. Firstly, the nonnegativity and boundedness of the solution for rumor-spreading model are proved by reductio ad absurdum. Secondly, both the basic reproduction number and the stability of the rumor-free equilibrium are systematically discussed. Whereafter, the global stability of rumor-prevailing equilibrium is explored by utilizing Lyapunov method and LaSalle’s invariance principle. Finally, the sensitivity analysis and the numerical simulation are respectively presented to analyze the impact of model parameters and illustrate the validity of theoretical results.

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