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.

scholarly journals A gradient recovery method based on an oblique projection and boundary modification

2017 ◽  
Vol 58 ◽  
pp. 34 ◽  
Author(s):  
Muhammad Ilyas ◽  
Bishnu P. Lamichhane ◽  
Michael H. Meylan
Keyword(s):  
Oblique Projection ◽  
Gradient Recovery ◽  
Recovery Method ◽  
Boundary Modification

scholarly journals A $C^1$ Virtual Element Method for the Cahn--Hilliard Equation with Polygonal Meshes

2016 ◽  
Vol 54 (1) ◽  
pp. 34-56 ◽  
Author(s):  
P. F. Antonietti ◽  
L. Beira͂o da Veiga ◽  
S. Scacchi ◽  
M. Verani
Keyword(s):  
Virtual Element Method ◽  
Polygonal Meshes ◽  
Hilliard Equation ◽  
Virtual Element ◽  
Cahn Hilliard Equation ◽  
Element Method

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 Superconvergent gradient recovery for virtual element methods

2019 ◽  
Vol 29 (11) ◽  
pp. 2007-2031 ◽  
Author(s):  
Hailong Guo ◽  
Cong Xie ◽  
Ren Zhao
Keyword(s):  
Posteriori Error ◽  
Error Estimator ◽  
A Posteriori ◽  
Gradient Recovery ◽  
A Posteriori Error ◽  
Error Estimators ◽  
Virtual Element Methods ◽  
Virtual Element ◽  
Posteriori Error Estimators ◽  
Element Method

Virtual element method is a new promising finite element method using general polygonal meshes. Its optimal a priori error estimates are well established in the literature. In this paper, we take a different viewpoint. We try to uncover the superconvergent property of virtual element methods by doing some local post-processing only on the degrees of freedom. Using the linear virtual element method as an example, we propose a universal gradient recovery procedure to improve the accuracy of gradient approximation for numerical methods using general polygonal meshes. Its capability of serving as a posteriori error estimators in adaptive computation is also investigated. Compared to the existing residual-type a posteriori error estimators for the virtual element methods, the recovery-type a posteriori error estimator based on the proposed gradient recovery technique is much simpler in implementation and it is asymptotically exact. A series of benchmark tests are presented to numerically illustrate the superconvergence of recovered gradient and validate the asymptotic exactness of the recovery-based a posteriori error estimator.

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