Virtual Element Method for the Laplace-Beltrami equation on surfaces

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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Vertex Displacement-Based Discontinuous Deformation Analysis Using Virtual Element Method

Materials ◽

10.3390/ma14051252 ◽

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.

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An H1-conforming virtual element for Darcy and Brinkman equations

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202518500057 ◽

2017 ◽

Vol 28 (01) ◽

pp. 159-194 ◽

Cited By ~ 32

Author(s):

Giuseppe Vacca

Keyword(s):

Stokes Equations ◽

Brinkman Equation ◽

Optimal Order ◽

Virtual Element Method ◽

Element Space ◽

Optimal Order Of Convergence ◽

Brinkman Equations ◽

Divergence Free ◽

Virtual Element ◽

Element Method

The focus of this paper is on developing a virtual element method (VEM) for Darcy and Brinkman equations. In [L. Beirão da Veiga, C. Lovadina and G. Vacca, ESAIM Math. Model. Numer. Anal. 51 (2017)], we presented a family of virtual elements for Stokes equations and we defined a new virtual element space of velocities such that the associated discrete kernel is pointwise divergence-free. We use a slightly different virtual element space having two fundamental properties: the [Formula: see text]-projection onto [Formula: see text] is exactly computable on the basis of the degrees of freedom, and the associated discrete kernel is still pointwise divergence-free. The resulting numerical scheme for the Darcy equation has optimal order of convergence and [Formula: see text]-conforming velocity solution. We can apply the same approach to develop a robust virtual element method for the Brinkman equation that is stable for both the Stokes and Darcy limit case. We provide a rigorous error analysis of the method and several numerical tests.

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Numerical experiments with the extended virtual element method for the Laplace problem with strong discontinuities

10.2172/1434461 ◽

2018 ◽

Author(s):

Gianmarco Manzini ◽

Elena Benvenuti ◽

Andrea Chiozzi ◽

N. Sukumar

Keyword(s):

Numerical Experiments ◽

Virtual Element Method ◽

Strong Discontinuities ◽

Virtual Element ◽

Laplace Problem ◽

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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