Virtual Element Method for general second-order elliptic problems on polygonal meshes

2016 ◽  
Vol 26 (04) ◽  
pp. 729-750 ◽  
Author(s):  
L. Beirão da Veiga ◽  
F. Brezzi ◽  
L. D. Marini ◽  
A. Russo
Keyword(s):  
Variable Coefficients ◽  
Second Order ◽  
Projection Operators ◽  
Virtual Element Method ◽  
Basic Principles ◽  
Order Elliptic Operator ◽  
Smooth Coefficients ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
Element Method

We consider the discretization of a boundary value problem for a general linear second-order elliptic operator with smooth coefficients using the Virtual Element approach. As in [A. H. Schatz, An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comput. 28 (1974) 959–962] the problem is supposed to have a unique solution, but the associated bilinear form is not supposed to be coercive. Contrary to what was previously done for Virtual Element Methods (as for instance in [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]), we use here, in a systematic way, the [Formula: see text]-projection operators as designed 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]. In particular, the present method does not reduce to the original Virtual Element Method of [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] for simpler problems as the classical Laplace operator (apart from the lowest-order cases). Numerical experiments show the accuracy and the robustness of the method, and they show as well that a simple-minded extension of the method in [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 case of variable coefficients produces, in general, sub-optimal results.

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 A gradient recovery method based on an oblique projection for the virtual element method

2019 ◽  
Vol 60 ◽  
pp. C201-C214
Author(s):  
Balaje Kalyanaraman ◽  
Bishnu Lamichhane ◽  
Michael Meylan
Keyword(s):  
Basis Functions ◽  
Oblique Projection ◽  
Gradient Recovery ◽  
Virtual Element Method ◽  
Polygonal Meshes ◽  
Recovery Method ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
Boundary Modification ◽  
Element Method

The virtual element method is an extension of the finite element method on polygonal meshes. The virtual element basis functions are generally unknown inside an element and suitable projections of the basis functions onto polynomial spaces are used to construct the elemental stiffness and mass matrices. We present a gradient recovery method based on an oblique projection, where the gradient of the L2-polynomial projection of a solution is projected onto a virtual element space. This results in a computationally efficient numerical method. We present numerical results computing the gradients on different polygonal meshes to demonstrate the flexibility of the method. References B. Ahmad, A. Alsaedi, F. Brezzi, L. D. Marini, and A. Russo. Equivalent projectors for virtual element methods. Comput. Math. Appl., 66(3):376391, 2013. doi:10.1016/j.camwa.2013.05.015. L. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Manzini, L. D. Marini, and A. Russo. Basic principles of virtual element methods. Math. Mod. Meth. Appl. Sci., 23(01): 199214, 2013. doi:10.1142/S0218202512500492. L. Beirao da Veiga, F. Brezzi, L. D. Marini, and A. Russo. The hitchhiker's guide to the virtual element method. Math. Mod. Meth. Appl. Sci., 24(08): 15411573, 2014. doi:10.1142/S021820251440003X. Ilyas, M. and Lamichhane, B. P. and Meylan, M. H. A gradient recovery method based on an oblique projection and boundary modification. In Proceedings of the 18th Biennial Computational Techniques and Applications Conference, CTAC-2016, volume 58 of ANZIAM J., pages C34C45, 2017. doi:10.21914/anziamj.v58i0.11730. B. P. Lamichhane. A gradient recovery operator based on an oblique projection. Electron. Trans. Numer. Anal., 37:166172, 2010. URL http://etna.mcs.kent.edu/volumes/2001-2010/vol37/abstract.php?vol=37&pages=166-172. O. J. Sutton. Virtual element methods. PhD thesis, University of Leicester, Department of Mathematics, 2017. URL http://hdl.handle.net/2381/39955. C. Talischi, G. H. Paulino, A. Pereira, and I. F. M. Menezes. Polymesher: a general-purpose mesh generator for polygonal elements written in matlab. Struct. Multidiscip. O., 45(3):309328, 2012. doi:10.1007/s00158-011-0706-z. G. Vacca and L. Beirao da Veiga. Virtual element methods for parabolic problems on polygonal meshes. Numer. Meth. Part. D. E., 31(6): 21102134, 2015. doi:10.1002/num.21982. J. Xu and Z. Zhang. Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comput., 73:11391152, 2004. doi:10.1090/S0025-5718-03-01600-4.

The Hitchhiker's Guide to the Virtual Element Method

2014 ◽  
Vol 24 (08) ◽  
pp. 1541-1573 ◽  
Author(s):  
L. Beirão da Veiga ◽  
F. Brezzi ◽  
L. D. Marini ◽  
A. Russo
Keyword(s):  
Second Order ◽  
Computer Implementation ◽  
Virtual Element Method ◽  
Order Problem ◽  
Virtual Element ◽  
Element Method

We present the essential ingredients in the Virtual Element Method for a simple linear elliptic second-order problem. We emphasize its computer implementation, which will enable interested readers to readily implement the method.

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.

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

2018 ◽  
Vol 77 (3) ◽  
pp. 1874-1908 ◽  
Author(s):  
Lorenzo Mascotto ◽  
Ilaria Perugia ◽  
Alexander Pichler
Keyword(s):  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

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.

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