Nonconforming Virtual Element Method Research Articles

The stabilized nonconforming virtual element method for the Darcy–Stokes problem

A stabilized nonconforming virtual element method is designed in order to solve the Darcy–Stokes problem, which preserves a divergence-free approximation to the velocity. The same degrees of freedom as the usual Crouzeix–Raviart-type virtual element is used, but a different virtual element space is obtained by modifying the conforming Stokes virtual element with the H1-projection operator. The proposed stabilized scheme contains two jump penalty terms over edges. One is the penalty for jumps of velocity approximation and the other one is the penalty for jumps of its normal component. We analyze this method’s well-posedness and prove its uniform convergence in a discrete energy norm. Finally, we verify the validity of this stabilized scheme by some numerical experiments.

A C0 Nonconforming Virtual Element Method for the Kirchhoff Plate Obstacle Problem

This paper investigates a novel C0 nonconforming virtual element method (VEM) for solving the Kirchhoff plate obstacle problem, which is described by a fourth-order variational inequality (VI) of the first kind. In our study, we distinguish our approach by introducing new internal degrees of freedom to the traditional lowest-order C0 nonconforming VEM, which originally lacked such degrees. This addition not only facilitates error estimation but also enhances its intuitiveness. Importantly, our novel C0 nonconforming VEM naturally satisfies the constraints of the obstacle problem. We then establish an a priori error estimate for our novel C0 nonconforming VEM, with the result indicating that the lowest order of our method achieves optimal convergence. Finally, we present a numerical example to validate the theoretical result.

A parameter robust reconstruction nonconforming virtual element method for the incompressible poroelasticity model

A reconstruction nonconforming virtual element method for the incompressible poroelasticity model is developed and analyzed. We investigate to determine the divergence-free displacement in incompressible poroelasticity model on polygons. The presented method mainly involves the parameter robust for μ also can be considered as the real pressure robustness for the virtual element. Using the H(div) space on the polygons to build the reconstruction operator, the method can be applied on general polygonal meshes, satisfy pressure robustness and overcome Poisson locking in incompressible model. The results are corroborated by theoretical derivations as well as numerical results.

Nonconforming virtual element methods for velocity-pressure Stokes eigenvalue problem

In this work, we investigate a divergence-free, nonconforming virtual element method for the numerical approximation of the Stokes eigenvalue problems using polygonal unstructured meshes on polygonal domain. By employing an elliptic projection operator, we “enhance” the definition of the virtual element space to make the L2-projection operator computable with optimal order. Furthermore, we prove the optimal order of convergence of the eigenfunction approximation and the double order of convergence of the eigenvalue approximation through compactness arguments for the solution operator (Babuška-Osborn Theory). Finally, we present a set of numerical experiments to assess the good approximation behavior of the method and show its computational performance.

The nonconforming virtual element method for optimal control problem governed by Stokes equations

The nonconforming virtual element method for optimal control problem governed by Stokes equations

An analysis of nonconforming virtual element methods on polytopal meshes with small faces

In this paper, we analyze nonconforming virtual element methods on polytopal meshes with small faces for the second-order elliptic problem. We propose new stability forms for 2D and 3D nonconforming virtual element methods. For the 2D case, the stability form is defined by the sum of an inner product of approximate tangential derivatives and a weighed [Formula: see text]-inner product of certain projections on the mesh element boundaries. For the 3D case, the stability form is defined by a weighted [Formula: see text]-inner product on the mesh element boundaries. We prove the optimal convergence of the nonconforming virtual element methods equipped with such stability forms. Finally, several numerical experiments are presented to verify our analysis and compare the performance of the proposed stability forms with the standard stability form [B. Ayuso de Dios, K. Lipnikov and G. Manzini, The nonconforming virtual element method, ESAIM Math. Model. Numer. Anal. 50 (2016) 879–904].

Nonconforming virtual element discretization for the transmission eigenvalue problem

A C0 nonconforming virtual element method (VEM) is employed for the discretization of nonselfadjoint fourth-order eigenvalue problems derived from the transmission eigenvalue problem on general polygonal meshes. Further, we linearize the model problem by introducing an auxiliary variable and thus the variational problem becomes double in size. We discretize the modified scheme by using suitable projection operators and acquire the convergence analysis under standard assumptions on the polygonal meshes. We establish that the resulting discretization provides a correct approximation of the spectrum and conclude the convergence of the eigenfunctions and eigenvalues. A variety of representative examples are examined to justify the theoretical estimates.

Dissipative nonconforming virtual element method for the fourth order nonlinear extended Fisher-Kolmogorov equation

A unified framework of nonconforming virtual element methods (VEMs), including C0 and fully nonconforming virtual elements, are built for the fourth order extended Fisher-Kolmogorov (EFK) equation. For the constructed semi-discrete and fully discrete schemes, the unique solvability is obtained. Then the numerical schemes are proved to be dissipative in the senses of discrete energy, which can be further used to derive the priori bounds of the discrete solutions. In addition, a new analytical technique is used to derive the unconditional convergence of the numerical schemes without any restriction on the grid ratio. Finally, numerical experiments are provided to confirm the theoretical results.

A virtual element method for overcoming locking phenomena in Biot’s consolidation model

A novel algorithm for the three-field formulation of Biot’s consolidation model based on mixed and divergence-free nonconforming virtual element methods is developed and analyzed. By establishing a discrete counterpart of Korn’s inequality, we ensure the well-posedness of this algorithm without special constraints in the context of nonconforming methods. In addition, we also derive a unified error estimate for this fully discrete algorithm no matter whether the specific storage coefficient vanishes or not. Moreover, this algorithm has several features, including supporting general polygonal meshes and arbitrary space approximation orders, and without Poisson’s locking and pressure oscillations. Numerical experiments are presented to validate the performance of this algorithm.

Conforming and nonconforming virtual element methods for fourth order nonlocal reaction diffusion equation

In this work, we have designed conforming and nonconforming virtual element methods (VEM) to approximate non-stationary nonlocal biharmonic equation on general shaped domain. By employing Faedo–Galerkin technique, we have proved the existence and uniqueness of the continuous weak formulation. Upon applying Brouwer’s fixed point theorem, the well-posedness of the fully discrete scheme is derived. Further, following [J. Huang and Y. Yu, A medius error analysis for nonconforming virtual element methods for Poisson and biharmonic equations, J. Comput. Appl. Math. 386 (2021) 113229], we have introduced Enrichment operator and derived a priori error estimates for fully discrete schemes on polygonal domains, not necessarily convex. The proposed error estimates are justified with some benchmark examples.

Variable-time-step BDF2 nonconforming VEM for coupled Ginzburg-Landau equations

In this paper, we solve the coupled Ginzburg-Landau equations by using a linearized variable-time-step second order backward differentiation formula in time combining with a nonconforming virtual element method in space. Based on the techniques of the discrete orthogonal convolution kernels and the discrete complementary convolution kernels, the unconditional optimal L2-norm error estimate of the fully discrete scheme was presented by using the error splitting technique under the mild restriction on the ratio of adjacent time-steps ratios. Finally, numerical experiments illustrate the correctness of our theoretical analysis.

A lowest-order locking-free nonconforming virtual element method based on the reduced integration technique for linear elasticity problems

We develop a lowest-order nonconforming virtual element method for planar linear elasticity, which can be viewed as an extension of the idea in Falk (1991) to the virtual element method (VEM), with the family of polygonal meshes satisfying a very general geometric assumption. The method is shown to be uniformly convergent for the nearly incompressible case with optimal rates of convergence. The crucial step is to establish the discrete Korn's inequality, yielding the coercivity of the discrete bilinear form. We also provide a unified locking-free scheme both for the conforming and nonconforming VEMs in the lowest-order case. Numerical results validate the feasibility and effectiveness of the proposed numerical algorithms.

The virtual element method for a nonhomogeneous double obstacle problem of Kirchhoff plate

This paper is intended to propose and analyze the lowest order conforming and nonconforming virtual element methods (VEMs) for a nonhomogeneous double obstacle problem in fourth-order variational inequality. We first present an abstract framework for the error analysis of an abstract discrete method. Then, we develop the conforming and fully nonconforming VEMs for the previous problem, with the optimal error estimates obtained by using the previous abstract framework combined with two modified interpolation operators and the related estimates. The discrete problem is solved by the primal–dual active set algorithm and the numerical results are in good agreement with theoretical findings.

Error Analysis of Nonconforming Virtual Element Method for Stokes Problem with Low Regularity

In this paper, the nonconforming virtual element method is used to solve the Stokes problem where the velocity and pressure are allowed to have the low regularity. With the help of an enriching operator, the consistency error is estimated under the low regularity condition. Then the optimal error estimates are obtained for the velocity and pressure approximations, which implies that the nonconforming virtual element method has the good convergence even for the Stokes problem with the low regularity.

Nonconforming virtual element methods for the fourth-order variational inequalities of the first kind

We study the C0 nonconforming and fully nonconforming virtual element methods (VEMs) for solving two Kirchhoff plate problems, which are fourth-order variational inequalities of the first kind. For the Kirchhoff plate obstacle problem, we propose a modified fully nonconforming virtual element method. A priori error estimations are established for both the C0 nonconforming and fully nonconforming VEMs for the two Kirchhoff plate problems, and we show that the lowest-order VEMs achieve optimal convergence order. Two numerical examples are presented to support the theoretical results.

The nonconforming virtual element method for semilinear elliptic problems

The nonconforming virtual element method (VEM) for the semilinear elliptic problem is developed in this paper. The nonlinear right-hand side is approximated by using the L2 projection. The optimal convergence of the nonconforming VEM in the broken H1 norm is proved. Finally, some numerical experiments are carried out to support the theoretical results.

A priori and a posteriori error analysis of the lowest-order NCVEM for second-order linear indefinite elliptic problems

The nonconforming virtual element method (NCVEM) for the approximation of the weak solution to a general linear second-order non-selfadjoint indefinite elliptic PDE in a polygonal domain Omega is analyzed under reduced elliptic regularity. The main tool in the a priori error analysis is the connection between the nonconforming virtual element space and the Sobolev space H^1_0(Omega ) by a right-inverse J of the interpolation operator I_h. The stability of the discrete solution allows for the proof of existence of a unique discrete solution, of a discrete inf-sup estimate and, consequently, for optimal error estimates in the H^1 and L^2 norms. The explicit residual-based a posteriori error estimate for the NCVEM is reliable and efficient up to the oscillation terms. Numerical experiments on different types of polygonal meshes illustrate the robustness of an error estimator and support the improved convergence rate of an adaptive mesh-refinement in comparison to the uniform mesh-refinement.

Mixed virtual volume methods for elliptic problems

We develop a class of mixed virtual volume methods for elliptic problems on polygonal/polyhedral grids. Unlike the mixed virtual element methods introduced in [22,13], our methods are reduced to symmetric, positive definite problems for the primary variable without using Lagrangian multipliers. We start from the usual way of changing the given equation into a mixed system using the Darcy's law, u=−K∇p. By integrating the system of equations with some judiciously chosen test spaces on each element, we define new mixed virtual volume methods of all orders. We show that these new schemes are equivalent to the nonconforming virtual element methods for the primal variable p.Once the primary variable is computed solving the symmetric, positive definite system, all the degrees of freedom for the Darcy velocity are locally computed. Also, the L2-projection onto the polynomial space is easy to compute. Hence our work opens an easy way to compute Darcy velocity on the polygonal/polyhedral grids. For the lowest order case, we give a formula to compute a Raviart-Thomas space like representation which satisfies the conservation law.An optimal error analysis is carried out and numerical results are presented which support the theory.

A formal construction of a divergence-free basis in the nonconforming virtual element method for the Stokes problem

We develop a formal construction of a pointwise divergence-free basis in the nonconforming virtual element method of arbitrary order for the Stokes problem introduced in Zhao et al. (SIAM J. Numer. Anal. 57(6):2730–2759, 2019). The proposed construction can be seen as a generalization of the divergence-free basis in Crouzeix-Raviart finite element space (Brenner, Math. Comp. 55(192):411–437, 1990; Thomasset, 1981) to the virtual element space. Using the divergence-free basis obtained from our construction, we can eliminate the pressure variable from the mixed system and obtain a symmetric positive definite system. Several numerical tests are presented to confirm the efficiency and the accuracy of our construction.

Lowest-order virtual element methods for linear elasticity problems

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 Hansbo and Larson (2003) 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. This approach can be seen as an extension of the idea of Kouhia and Stenberg suggested in Kouhia and Stenberg (1995) to the virtual element method. We show that our proposed methods satisfy the discrete Korn’s inequality. We also prove that the methods are convergent uniformly for the nearly incompressible case and the convergence rates are optimal.