Piecewise divergence-free nonconforming virtual elements are designed for Stokes problem in any dimensions. After introducing a local energy projector based on the Stokes problem and the stabilizat...

#2Rui Li(SNNU: Shaanxi Normal University)H-Index: 6

Last. Yufeng Nie(NPU: Northwestern Polytechnical University)

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Abstract In this paper, we propose and investigate a divergence-free reconstruction of the nonconforming virtual element for the Stokes problem. By constructing the computable Raviart–Thomas-like interpolation operator, we guarantee the independence between the velocity error estimation | u − u h | 1 , h and the continuous pressure p , as it happens for the divergence-free flow solver. Moreover, this modified scheme can also inherit the advantages of the classical nonconforming virtual element m...

#1Xuehai Huang(SUFE: Shanghai University of Finance and Economics)H-Index: 1

#1Xuehai Huang(SUFE: Shanghai University of Finance and Economics)H-Index: 9

Last. Xuehai Huang(SUFE: Shanghai University of Finance and Economics)

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The $H^m -nonconforming virtual elements of any order k on any shape of polytope in {\mathbb {R}}^n with constraints m> n and k\ge m are constructed in a universal way. A generalized Green’s identity for H^m inner product with m>n is derived, which is essential to devise the H^m -nonconforming virtual elements. By means of the local H^m projection and a stabilization term using only the boundary degrees of freedom, the H^m -nonconforming virtual element methods...

This paper has two objectives. On one side, we develop and test numerically divergence-free Virtual Elements in three dimensions, for variable “polynomial” order. These are the natural extension of...

#1Long Chen(UCI: University of California, Irvine)H-Index: 23

#2Xuehai Huang(SUFE: Shanghai University of Finance and Economics)H-Index: 9

A unified construction of the H^mnonconforming virtual elements of any order kis developed on any shape of polytope in \mathbb R^nwith constraints m\leq nand k\geq m As a vital tool in the construction, a generalized Green's identity for H^minner product is derived. The H^mnonconforming virtual element methods are then used to approximate solutions of the mharmonic equation. After establishing a bound on the jump related to the weak continuity, the optimal error estimate ...

We present the divergence-free nonconforming virtual element method for the Stokes problems. We first construct a nonconforming virtual element with continuous normal component and weak continuous ...

Some virtual element methods on polytopal meshes for the Stokes problem are proposed and analyzed. The pressure is approximated by discontinuous polynomials, while the velocity is discretized by H(div) virtual elements enriched with some tangential polynomials on the element boundaries. A weak symmetric gradient of the velocity is computed using the corresponding degree of freedoms. The main feature of the method is that it exactly preserves the divergence free constraint, and therefore the erro...

#2Li-Yeng Sung(LSU: Louisiana State University)H-Index: 23

We consider a model Poisson problem in ℝd (d = 2, 3) and establish error estimates for virtual element methods on polygonal or polyhedral meshes that can contain small edges (d = 2) or small faces (d = 3). Our results extend the ones in [L. Beirao 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. Beirao da Veiga, F. Brezzi, A. Cangiani, G. Man...

A family of virtual element methods for the two-dimensional Navier--Stokes equations is proposed and analyzed. The schemes provide a discrete velocity field which is pointwise divergence-free. A ri...

We present two kinds of lowest-order virtual element methods for planar linear elasticity problems. For the first one we use the nonconforming virtual element method with a stabilizing term. It can be interpreted as a modification of the nonconforming Crouzeix-Raviart finite element method as suggested in [22] to the virtual element method. For the second one we use the conforming virtual element for one component of the displacement vector and the nonconforming virtual element for the other. Th...

Last. Jianguo Huang(SJTU: Shanghai Jiao Tong University)H-Index: 28

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Abstract null null This paper is devoted to the numerical solution of a fourth-order elliptic variational inequality of the first kind by the virtual element method (VEM). The variational inequality models an obstacle problem for the Kirchhoff-Love plate. Both conforming and fully nonconforming VEMs are studied to solve the fourth-order elliptic variational inequality. Optimal order error estimates are derived in the discrete energy norm, under certain solution regularity assumptions. The primal...

One conforming and one non-conforming virtual element Hessian complexes on tetrahedral grids are constructed based on decompositions of polynomial tensor space. They are applied to discretize the linearized time-independent Einstein-Bianchi system.