A divergence-free weak virtual element method for the Navier-Stokes equation on polygonal meshes

2021 ◽  
Vol 47 (6) ◽  
Author(s):  
Gang Wang ◽  
Feng Wang ◽  
Yinnian He
Keyword(s):  
Stokes Equation ◽  
Navier Stokes ◽  
Virtual Element Method ◽  
Navier Stokes Equation ◽  
Polygonal Meshes ◽  
Divergence Free ◽  
Virtual Element ◽  
Element Method

A mixed virtual element method for the Navier–Stokes equations

2018 ◽  
Vol 28 (14) ◽  
pp. 2719-2762 ◽  
Author(s):  
Gabriel N. Gatica ◽  
Mauricio Munar ◽  
Filánder A. Sequeira
Keyword(s):  
Fixed Point ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Dirichlet Boundary Conditions ◽  
Navier Stokes ◽  
Equilibrium Equations ◽  
Virtual Element Method ◽  
Navier Stokes Equations ◽  
Virtual Element ◽  
Element Method

A mixed virtual element method (mixed-VEM) for a pseudostress-velocity formulation of the two-dimensional Navier–Stokes equations with Dirichlet boundary conditions is proposed and analyzed in this work. More precisely, we employ a dual-mixed approach based on the introduction of a nonlinear pseudostress linking the usual linear one for the Stokes equations and the convective term. In this way, the aforementioned new tensor together with the velocity constitute the only unknowns of the problem, whereas the pressure is computed via a postprocessing formula. In addition, the resulting continuous scheme is augmented with Galerkin type terms arising from the constitutive and equilibrium equations, and the Dirichlet boundary condition, all them multiplied by suitable stabilization parameters, so that the Banach fixed-point and Lax–Milgram theorems are applied to conclude the well-posedness of the continuous and discrete formulations. Next, we describe the main VEM ingredients that are required for our discrete analysis, which, besides projectors commonly utilized for related models, include, as the main novelty, the simultaneous use of virtual element subspaces for [Formula: see text] and [Formula: see text] in order to approximate the velocity and the pseudostress, respectively. Then, the discrete bilinear and trilinear forms involved, their main properties and the associated mixed virtual scheme are defined, and the corresponding solvability analysis is performed using again appropriate fixed-point arguments. Moreover, Strang-type estimates are applied to derive the a priori error estimates for the two components of the virtual element solution as well as for the fully computable projections of them and the postprocessed pressure. As a consequence, the corresponding rates of convergence are also established. Finally, we follow the same approach employed in previous works by some of the authors and introduce an element-by-element postprocessing formula for the fully computable pseudostress, thus yielding an optimally convergent approximation of this unknown with respect to the broken [Formula: see text]-norm.

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

scholarly journals Numerical solution of Navier stokes equation using control volume and finite element method

2016 ◽  
Vol 5 (1) ◽  
pp. 63
Author(s):  
Musa Adam Aigo
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Numerical Solution ◽  
Stokes Equation ◽  
Control Volume ◽  
Navier Stokes ◽  
Weak Formulation ◽  
Navier Stokes Equation ◽  
Fast Solution ◽  
Element Method

<p>The aim of this paper is twofold first we will  provide a numerical solution of the Navier Stokes equation using the Projection technique and finite element method. The problem will be introduced in weak formulation and a Finite Element method will be developed, then solve in a fast way the sparse system derived. Second, the projection method with Control volume approach will be applied to get a fast solution, in iterations count.</p>

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.

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