The mixed virtual element method on curved edges in two dimensions

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.

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On the choice of the internal degrees of freedom for the nodal Virtual Element Method in two dimensions

Computers & Mathematics with Applications ◽

10.1016/j.camwa.2016.03.016 ◽

2016 ◽

Vol 72 (8) ◽

pp. 1968-1976 ◽

Cited By ~ 6

Author(s):

Alessandro Russo

Keyword(s):

Degrees Of Freedom ◽

Two Dimensions ◽

Virtual Element Method ◽

Virtual Element ◽

Internal Degrees Of Freedom ◽

Element Method

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

CALCOLO ◽

10.1007/s10092-021-00418-5 ◽

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.

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Virtual Element Method for Solving an Inhomogeneous Brusselator Model With and Without Cross-Diffusion in Pattern Formation

Journal of Scientific Computing ◽

10.1007/s10915-021-01626-5 ◽

2021 ◽

Vol 89 (1) ◽

Author(s):

Mehdi Dehghan ◽

Zeinab Gharibi

Keyword(s):

Pattern Formation ◽

Virtual Element Method ◽

Cross Diffusion ◽

Brusselator Model ◽

Virtual Element ◽

Element Method

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Non-conforming Harmonic Virtual Element Method: $$h$$ h - and $$p$$ p -Versions

Journal of Scientific Computing ◽

10.1007/s10915-018-0797-4 ◽

2018 ◽

Vol 77 (3) ◽

pp. 1874-1908 ◽

Cited By ~ 9

Author(s):

Lorenzo Mascotto ◽

Ilaria Perugia ◽

Alexander Pichler

Keyword(s):

Virtual Element Method ◽

Virtual Element ◽

Element Method

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The virtual element method for resistive magnetohydrodynamics

Computer Methods in Applied Mechanics and Engineering ◽

10.1016/j.cma.2021.113815 ◽

2021 ◽

Vol 381 ◽

pp. 113815

Author(s):

S. Naranjo Alvarez ◽

V. Bokil ◽

V. Gyrya ◽

G. Manzini

Keyword(s):

Virtual Element Method ◽

Virtual Element ◽

Element Method

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A virtual element method for the miscible displacement of incompressible fluids in porous media

Computer Methods in Applied Mechanics and Engineering ◽

10.1016/j.cma.2020.113649 ◽

2021 ◽

Vol 375 ◽

pp. 113649

Author(s):

L. Beirão da Veiga ◽

A. Pichler ◽

G. Vacca

Keyword(s):

Porous Media ◽

Incompressible Fluids ◽

Miscible Displacement ◽

Virtual Element Method ◽

Virtual Element ◽

Element Method

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Virtual Element Method for the Laplace-Beltrami equation on surfaces

ESAIM Mathematical Modelling and Numerical Analysis ◽

10.1051/m2an/2017040 ◽

2018 ◽

Vol 52 (3) ◽

pp. 965-993 ◽

Cited By ~ 8

Author(s):

Massimo Frittelli ◽

Ivonne Sgura

Keyword(s):

Numerical Experiments ◽

Beltrami Equation ◽

Convergence Result ◽

Point Of View ◽

Polygonal Approximation ◽

Virtual Element Method ◽

Element Space ◽

First Order ◽

Virtual Element ◽

Element Method

We present and analyze a Virtual Element Method (VEM) for the Laplace-Beltrami equation on a surface in ℝ3, that we call Surface Virtual Element Method (SVEM). The method combines the Surface Finite Element Method (SFEM) (Dziuk, Eliott, G. Dziuk and C.M. Elliott., Acta Numer. 22 (2013) 289–396.) and the recent VEM (Beirão da Veiga et al., Math. Mod. Methods Appl. Sci. 23 (2013) 199–214.) in order to allow for a general polygonal approximation of the surface. We account for the error arising from the geometry approximation and in the case of polynomial order k = 1 we extend to surfaces the error estimates for the interpolation in the virtual element space. We prove existence, uniqueness and first order H1 convergence of the numerical solution.We highlight the differences between SVEM and VEM from the implementation point of view. Moreover, we show that the capability of SVEM of handling nonconforming and discontinuous meshes can be exploited in the case of surface pasting. We provide some numerical experiments to confirm the convergence result and to show an application of mesh pasting.

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