scholarly journals p- and hp- virtual elements for the Stokes problem

2021 ◽  
Vol 47 (2) ◽  
Author(s):  
A. Chernov ◽  
C. Marcati ◽  
L. Mascotto
Keyword(s):  
Exponential Convergence ◽  
Stokes Problem ◽  
Virtual Element Method ◽  
Polygonal Domains ◽  
Numerical Tests ◽  
Right Hand ◽  
Stokes Problems ◽  
Explicit Analysis ◽  
Virtual Element ◽  
Numer Math

AbstractWe analyse the p- and hp-versions of the virtual element method (VEM) for the Stokes problem on polygonal domains. The key tool in the analysis is the existence of a bijection between Poisson-like and Stokes-like VE spaces for the velocities. This allows us to re-interpret the standard VEM for Stokes as a VEM, where the test and trial discrete velocities are sought in Poisson-like VE spaces. The upside of this fact is that we inherit from Beirão da Veiga et al. (Numer. Math. 138(3), 581–613, 2018) an explicit analysis of best interpolation results in VE spaces, as well as stabilization estimates that are explicit in terms of the degree of accuracy p of the method. We prove exponential convergence of the hp-VEM for Stokes problems with regular right-hand sides. We corroborate the theoretical estimates with numerical tests for both the p- and hp-versions of the method.

scholarly journals The harmonic virtual element method: stabilization and exponential convergence for the Laplace problem on polygonal domains

2018 ◽  
Vol 39 (4) ◽  
pp. 1787-1817 ◽  
Author(s):  
Alexey Chernov ◽  
Lorenzo Mascotto
Keyword(s):  
Finite Element ◽  
Laplace Equation ◽  
Degrees Of Freedom ◽  
Exponential Convergence ◽  
Harmonic Polynomial ◽  
Harmonic Polynomials ◽  
Virtual Element Method ◽  
Polygonal Domains ◽  
Virtual Element ◽  
Element Method

Abstract We introduce the harmonic virtual element method (VEM) (harmonic VEM), a modification of the VEM (Beirão da Veiga et al. (2013) Basic principles of virtual element methods. Math. Models Methods Appl. Sci., 23, 199–214.) for the approximation of the two-dimensional Laplace equation using polygonal meshes. The main difference between the harmonic VEM and the VEM is that in the former method only boundary degrees of freedom are employed. Such degrees of freedom suffice for the construction of a proper energy projector on (piecewise harmonic) polynomial spaces. The harmonic VEM can also be regarded as an ‘$H^1$-conformisation’ of the Trefftz discontinuous Galerkin-finite element method (TDG-FEM) (Hiptmair et al. (2014) Approximation by harmonic polynomials in starshaped domains and exponential convergence of Trefftz hp-DGFEM. ESAIM Math. Model. Numer. Anal., 48, 727–752.). We address the stabilization of the proposed method and develop an hp version of harmonic VEM for the Laplace equation on polygonal domains. As in TDG-FEM, the asymptotic convergence rate of harmonic VEM is exponential and reaches order $\mathscr{O}(\exp (-b\sqrt [2]{N}))$, where $N$ is the number of degrees of freedom. This result overperforms its counterparts in the framework of hp FEM (Schwab, C. (1998)p- and hp-Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics. Clarendon Press Oxford.) and hp VEM (Beirão da Veiga et al. (2018) Exponential convergence of the hp virtual element method with corner singularity. Numer. Math., 138, 581–613.), where the asymptotic rate of convergence is of order $\mathscr{O}(\exp(-b\sqrt [3]{N}))$.

scholarly journals Exponential convergence of the hp virtual element method in presence of corner singularities

2017 ◽  
Vol 138 (3) ◽  
pp. 581-613 ◽  
Author(s):  
L. Beirão da Veiga ◽  
A. Chernov ◽  
L. Mascotto ◽  
A. Russo
Keyword(s):  
Exponential Convergence ◽  
Corner Singularities ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals Divergence-preserving reconstructions on polygons and a really pressure-robust virtual element method for the Stokes problem

2020 ◽  
Author(s):  
Derk Frerichs ◽  
Christian Merdon
Keyword(s):  
Stokes Problem ◽  
Two Dimensions ◽  
Momentum Balance ◽  
Test Functions ◽  
Virtual Test ◽  
Right Hand ◽  
Virtual Element Methods ◽  
Divergence Free ◽  
Virtual Element ◽  
The Right

Abstract Nondivergence-free discretizations for the incompressible Stokes problem may suffer from a lack of pressure-robustness characterized by large discretizations errors due to irrotational forces in the momentum balance. This paper argues that also divergence-free virtual element methods on polygonal meshes are not really pressure-robust as long as the right-hand side is not discretized in a careful manner. To be able to evaluate the right-hand side for the test functions, some explicit interpolation of the virtual test functions is needed that can be evaluated pointwise everywhere. The standard discretization via an $L^2$-best approximation does not preserve the divergence, and so destroys the orthogonality between divergence-free test functions and possibly eminent gradient forces in the right-hand side. To repair this orthogonality and restore pressure-robustness, another divergence-preserving reconstruction is suggested based on Raviart–Thomas approximations on local subtriangulations of the polygons. All findings are proven theoretically and are demonstrated numerically in two dimensions. The construction is also interesting for hybrid high-order methods on polygonal or polyhedral meshes.

The nonconforming virtual element method for the Darcy–Stokes problem

2020 ◽  
Vol 370 ◽  
pp. 113251
Author(s):  
Jikun Zhao ◽  
Bei Zhang ◽  
Shipeng Mao ◽  
Shaochun Chen
Keyword(s):  
Stokes Problem ◽  
Virtual Element Method ◽  
Virtual Element ◽  
Element Method

scholarly journals The Stokes Complex for Virtual Elements with Application to Navier–Stokes Flows

2019 ◽  
Vol 81 (2) ◽  
pp. 990-1018 ◽  
Author(s):  
L. Beirão da Veiga ◽  
D. Mora ◽  
G. Vacca
Keyword(s):  
Degrees Of Freedom ◽  
Differential Operators ◽  
Complex Structure ◽  
Navier Stokes ◽  
Virtual Element Method ◽  
Element Space ◽  
Trilinear Form ◽  
Numerical Tests ◽  
Virtual Element ◽  
Associated Differential Operators

Abstract In the present paper, we investigate the underlying Stokes complex structure of the Virtual Element Method for Stokes and Navier–Stokes introduced in previous papers by the same authors, restricting our attention to the two dimensional case. We introduce a Virtual Element space $${\varPhi }_h \subset H^2({\varOmega })$$ Φ h ⊂ H 2 ( Ω ) and prove that the triad $$\{{\varPhi }_h, {\varvec{V}}_h, Q_h\}$$ { Φ h , V h , Q h } (with $${\varvec{V}}_h$$ V h and $$Q_h$$ Q h denoting the discrete velocity and pressure spaces) is an exact Stokes complex. Furthermore, we show the computability of the associated differential operators in terms of the adopted degrees of freedom and explore also a different discretization of the convective trilinear form. The theoretical findings are supported by numerical tests.

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