scholarly journals An H1-conforming virtual element for Darcy and Brinkman equations

2017 ◽  
Vol 28 (01) ◽  
pp. 159-194 ◽  
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.

scholarly journals Virtual Element Method for the Laplace-Beltrami equation on surfaces

2018 ◽  
Vol 52 (3) ◽  
pp. 965-993 ◽  
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.

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 virtual element method for a nonlocal FitzHugh–Nagumo model of cardiac electrophysiology

2019 ◽  
Vol 40 (2) ◽  
pp. 1544-1576 ◽  
Author(s):  
Verónica Anaya ◽  
Mostafa Bendahmane ◽  
David Mora ◽  
Mauricio Sepúlveda
Keyword(s):  
Cardiac Electrophysiology ◽  
A Priori Estimates ◽  
A Priori ◽  
Reaction Diffusion ◽  
Optimal Order ◽  
Computational Domain ◽  
Virtual Element Method ◽  
Compactness Criterion ◽  
Virtual Element ◽  
Element Method

AbstractWe present a virtual element method (VEM) for a nonlocal reaction–diffusion system of the cardiac electric field. For this system, we analyze an $H^1$-conforming discretization by means of VEM that can make use of general polygonal meshes. Under standard assumptions on the computational domain, we establish the convergence of the discrete solution by considering a series of a priori estimates and by using a general $L^p$ compactness criterion. Moreover, we obtain optimal order space-time error estimates in the $L^2$ norm. Finally, we report some numerical tests supporting the theoretical results.

scholarly journals Virtual element method for quasilinear elliptic problems

2019 ◽  
Vol 40 (4) ◽  
pp. 2450-2472 ◽  
Author(s):  
A Cangiani ◽  
P Chatzipantelidis ◽  
G Diwan ◽  
E H Georgoulis
Keyword(s):  
A Priori ◽  
Quasilinear Equation ◽  
Nonlinear Coefficient ◽  
Optimal Order ◽  
Discrete Problem ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Fixed Point Iterations ◽  
Quasilinear Elliptic ◽  
Element Method

Abstract A virtual element method for the quasilinear equation $-\textrm{div} ({\boldsymbol \kappa }(u)\operatorname{grad} u)=f$ using general polygonal and polyhedral meshes is presented and analysed. The nonlinear coefficient is evaluated with the piecewise polynomial projection of the virtual element ansatz. Well posedness of the discrete problem and optimal-order a priori error estimates in the $H^1$- and $L^2$-norm are proven. In addition, the convergence of fixed-point iterations for the resulting nonlinear system is established. Numerical tests confirm the optimal convergence properties of the method on general meshes.

scholarly journals The nonconforming Virtual Element Method for eigenvalue problems

2019 ◽  
Vol 53 (3) ◽  
pp. 749-774 ◽  
Author(s):  
Francesca Gardini ◽  
Gianmarco Manzini ◽  
Giuseppe Vacca
Keyword(s):  
Eigenvalue Problems ◽  
Inner Product ◽  
Optimal Order ◽  
Large Set ◽  
Discrete Problem ◽  
Virtual Element Method ◽  
Discrete Eigenvalue ◽  
Order Error ◽  
Virtual Element ◽  
Element Method

We analyse the nonconforming Virtual Element Method (VEM) for the approximation of elliptic eigenvalue problems. The nonconforming VEM allows to treat in the same formulation the two- and three-dimensional case. We present two possible formulations of the discrete problem, derived respectively by the nonstabilized and stabilized approximation of theL2-inner product, and we study the convergence properties of the corresponding discrete eigenvalue problem. The proposed schemes provide a correct approximation of the spectrum, in particular we prove optimal-order error estimates for the eigenfunctions and the usual double order of convergence of the eigenvalues. Finally we show a large set of numerical tests supporting the theoretical results, including a comparison with the conforming Virtual Element choice.

scholarly journals The NonConforming Virtual Element Method for the Stokes Equations

2016 ◽  
Vol 54 (6) ◽  
pp. 3411-3435 ◽  
Author(s):  
Andrea Cangiani ◽  
Vitaliy Gyrya ◽  
Gianmarco Manzini
Keyword(s):  
Stokes Equations ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals Vertex Displacement-Based Discontinuous Deformation Analysis Using Virtual Element Method

Materials ◽  
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.

Sign in / Sign up

Export Citation Format

Share Document