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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.

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 tangential component by enriching the previous H(div)-conforming virtual 2 element with some divergence-free functions from the C⁰-continuous H²-nonconforming virtual element. By imposing a restriction on each edge for the resulting nonconforming virtual element, we obtain the desired nonconforming virtual element with the less space dimension. The nonconforming virtual element provides the exact divergence-free approximation to the velocity and is proved to be convergent with the optimal convergence rate. Further, we present two exact sequences of differential complex between the H¹-nonconforming and H²-nonconforming virtual elements. Finally, the numerical results are shown to confirm the convergence of the nonconforming virtual element.

In this paper, we consider the nonconforming virtual element method (VEM) for the approximation of the time fractional reaction–subdiffusion equation involving the Caputo fractional derivative. For the numerical discrete method of the Caputo fractional derivative, we permit the use of nonuniform time steps, since they are helpful to deal with the non-smooth system. Meanwhile, the nonconforming VEM, which is constructed for any order of accuracy and for very general shaped polygonal and polyhedral meshes, is adopted for the discretization of the spatial direction. By introducing a new Ritz projection operator and using two extended ty L 2 -normpes of continuous and discrete fractional Grönwall inequalities, the optimal error estimates for the spatial semi-discrete and temporal-spatial fully discrete systems are proved detailedly. Besides, the fully discrete scheme is proved to be unconditionally stable with regard to the L 2 - and H 1 -norms, respectively. Finally, some numerical calculations are implemented to verify the theoretical results.

We propose a simple nonconforming virtual element for plate bending problems, which has few local degrees of freedom and provides the optimal convergence in H 2 -norm. Moreover, we prove the optimal error estimates in H 1 - and L 2 -norm. The nonconforming virtual element is constructed for any order of accuracy, but not C 0 -continuous. It is worth mentioning that, for the lowest-order case on triangular meshes the simplified nonconforming virtual element coincides with the well-known Morley element, so it can be taken as the extension of the Morley element to polygonal meshes. Finally, we verify the optimal convergence in H 2 -norm for the nonconforming virtual element by some numerical tests.

In this paper, we estimate the constants in the inverse inequalities for the finite ele- ment functions. Furthermore, we obtain the least upper bounds of the constants in inverse inequalities for the low-order finite element functions. Such explicit estimates of the con- stants can be used as computable error bounds for the finite element method.

We present the stabilized nonconforming virtual element method for linear elasticity problem in two dimensions. The jump penalty term is introduced to guarantee the stability of the discrete formulation as the stabilization term, which is obtained based on the discrete Korn’s inequality. In order to obtain the computability of jump penalty term, we reconstruct the lowest-order nonconforming virtual element by imposing some restrictions on the conforming virtual element space of order 2. We prove the interpolation error estimate for the virtual element and the ellipticity of the discrete bilinear form, so the resulting stabilized method is well-posed. Then we show the optimal convergence in the L 2 and H 1 norms. Moreover, this method is locking-free, i.e. the convergence is uniform with respect to the Lamé constant. Numerical results are provided to confirm the theoretical results.

Pore water pressure and the changes of crustal deformation, fault rupture and seismic activity has important influence. So the pore water pressure and load rock stress – hydro-mechanical coupling mechanism is very important. This paper mainly studies the rock specimens of hydraulic crack damage simulation. This study found: with the increase of the axial compression, sample is on the surface crack. Crack characteristics is smooth and continuously expanding. With the load increasing at the same time, the number of samples is also increased damage elements. The sample was through the cracks. This is due to the effect of water pressure to reduce the size of confining pressure .From the failure mechanism analysis, the distribution of stress non-uniform material will not uniformity, reflected in the actual rock because of the grain and the defects of the random distribution. When the load, the composition of force transmission effect of different deformation and stress in rock, the internal non-uniform stress concentration, local, it will directly cause the weak part, and micro cracks generated change the failure mode of materials.

We present a Hermite-type virtual element method with interior penalty to solve the fourth-order elliptic problem over general polygonal meshes, where some interior penalty terms are added to impose the C 1 continuity. A C 0 -continuous Hermite-type virtual element with local H 2 regularity is constructed, such that it can be used in the interior penalty scheme. We prove the boundedness of basis functions and interpolation error estimates of Hermite-type virtual element. After introducing a discrete energy norm, we present the optimal convergence of the interior penalty scheme. Compared with some existing methods, the proposed interior penalty method uses fewer degrees of freedom. Finally, we verify the theoretical results through some numerical examples.

Based on equilibration of side fluxes, an a posteriori error estimator is obtained for the linear triangular element for the Poisson equation, which can be computed locally. We present a procedure for constructing the estimator in which we use the Lagrange multiplier similar to the usual equilibrated residual method introduced by Ainsworth and Oden. The estimator is shown to provide guaranteed upper bound, and local lower bounds on the error up to a multiplicative constant depending only on the geometry. Based on this, we give another error estimator which can be directly constructed without solving local Neumann problems and also provide the two-sided bounds on the error. Finally, numerical tests show our error estimators are very efficient.

We consider residual-based a posteriori error estimation for lowest-order nonconforming finite element approximations of streamline-diffusion type for solving convection-diffusion problems. The resulting error estimator is semi-robust in the sense that it yields lower and upper bounds of the error which differ by a factor equal at most to the square root of the Péclet number. The error analysis is also shown to be applied to nonconforming finite element methods with face penalty and subgrid viscosity. Numerical results show that the estimator can be used to construct adaptive meshes.