Superconvergence of discontinuous Galerkin finite element method for the stationary Navier‐Stokes equations

This article focuses on discontinuous Galerkin method for the two‐ or three‐dimensional stationary incompressible Navier‐Stokes equations. The velocity field is approximated by discontinuous locally solenoidal finite element, and the pressure is approximated by the standard conforming finite element. Then, superconvergence of nonconforming finite element approximations is applied by using least‐squares surface fitting for the stationary Navier‐Stokes equations. The method ameliorates the two noticeable disadvantages about the given finite element pair. Finally, the superconvergence result is provided under some regular assumptions. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 421–436, 2007     

Lagrange interpolation and finite element superconvergence

We consider the finite element approximation of the Laplacian operator with the homogeneous Dirichlet boundary condition, and study the corresponding Lagrange interpolation in the context of finite element superconvergence. For d‐dimensional Q_k‐type elements with d ≥ 1 and k ≥ 1, we prove that the interpolation points must be the Lobatto points if the Lagrange interpolation and the finite element solution are superclose in H¹ norm. For d‐dimensional P_k‐type elements, we consider the standard Lagrange interpolation—the Lagrange interpolation with interpolation points being the principle lattice points of simplicial elements. We prove for d ≥ 2 and k ≥ d + 1 that such interpolation and the finite element solution are not superclose in both H¹ and L² norms and that not all such interpolation points are superconvergence points for the finite element approximation. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 33–59, 2004.     

Superconvergence analysis of finite element method for Poisson–Nernst–Planck equations

This article concerns with the superconvergence analysis of bilinear finite element method (FEM) for nonlinear Poisson–Nernst–Planck (PNP) equations. By employing high accuracy integral identities together with mean value technique, the superclose estimates in H¹‐norm are derived for the semi‐discrete and the backward Euler fully‐discrete schemes, which improve the suboptimal error estimate in L²‐norm in the previous literature. Furthermore, the global superconvergence results in H¹‐norm are obtained through interpolation postprocessing approach. Finally, a numerical example is provided to confirm the theoretical analysis.     

A note on a lowest order divergence‐free Stokes element on quadrilaterals

This short note reports a lowest order divergence‐free Stokes element on quadrilateral meshes. The velocity space is based on a P₁ spline element over the crisscross partition of a quadrilateral, and the pressure is approximated by piecewise constant. For a given quadrilateral mesh, this element is stable if and only if the well‐known Q₁‐P₀ element is also stable. Although this method is a subspace method of Qin‐Zhang's P₁‐P₀ element, their velocity solutions are precisely equal. Moreover, an explicit basis representation is also provided. These theoretical findings are verified by numerical tests.     

Stokes型积分-微分方程Q_2-P_1元的超收敛分析

本文研究Q2-P1混合元对Stokes型积分-微分方程的有限元方法.利用积分恒等式技巧给出了关于流体速度u和压力p的误差估计,特别是在压力p的误差中去掉了影响解的稳定性的1因子t-2,改善了以往文献的结果.同时,通过构造适当的插值后处理算子得到了整体超收敛结果.     

Superconvergence of a two‐grid method for fourth‐order dispersive‐dissipative wave equations

In this paper, the superconvergence analysis of a two‐grid method (TGM) with low‐order finite elements is presented for the fourth‐order dispersive‐dissipative wave equations for a second order fully discrete scheme. The superclose estimates in the H¹‐norm on the two grids are obtained by the combination technique of the interpolation and Ritz projection. Then, with the help of the interpolated postprocessing technique, the global superconvergence properties are deduced. Finally, numerical results are provided to show the performance of the proposed TGM for conforming bilinear element and nonconforming element, respectively. It shows that the TGM is an effective method to the problem considered of our paper compared with the traditional Galerkin finite element method (FEM).     

Superconvergence of the direct discontinuous Galerkin method for a time‐fractional initial‐boundary value problem

A time‐fractional reaction–diffusion initial‐boundary value problem with periodic boundary condition is considered on Q ? Ω × [0, T] , where Ω is the interval [0, l] . Typical solutions of such problem have a weak singularity at the initial time t = 0. The numerical method of the paper uses a direct discontinuous Galerkin (DDG) finite element method in space on a uniform mesh, with piecewise polynomials of degree k ≥ 2 . In the temporal direction we use the L1 approximation of the Caputo derivative on a suitably graded mesh. We prove that at each time level of the mesh, our L1‐DDG solution is superconvergent of order k + 2 in L²(Ω) to a particular projection of the exact solution. Moreover, the L1‐DDG solution achieves superconvergence of order (k + 2) in a discrete L²(Q) norm computed at the Lobatto points, and order (k + 1) superconvergence in a discrete H¹(Q) seminorm at the Gauss points; numerical results show that these estimates are sharp.     

A fully discrete nonlinear Galerkin method for the 3D Navier–Stokes equations

The purpose of this paper is twofold: (i) We show that the Fourier‐based Nonlinear Galerkin Method (NLGM) constructs suitable weak solutions to the periodic Navier–Stokes equations in three space dimensions provided the large scale/small scale cutoff is appropriately chosen. (ii) If smoothness is assumed, NLGM always outperforms the Galerkin method by a factor equal to 1 in the convergence order of the H ¹‐norm for the velocity and the L²‐norm for the pressure. This is a purely linear superconvergence effect resulting from standard elliptic regularity and holds independently of the nature of the boundary conditions (whether periodicity or no‐slip BC is enforced). © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Unconditional superconvergent analysis of a new mixed finite element method for Ginzburg–Landau equation

In this article, unconditional superconvergent analysis of a linearized fully discrete mixed finite element method is presented for a class of Ginzburg–Landau equation based on the bilinear element and zero‐order Nédélec's element pair (Q₁₁/Q₀₁ × Q₁₀). First, a time‐discrete system is introduced to split the error into temporal error and spatial error, and the corresponding error estimates are deduced rigorously. Second, the unconditional superclose and optimal estimate of order O(h² + τ) for u in H¹‐norm and p = ?u in L²‐norm are derived respectively without the restrictions on the ratio between h and τ, where h is the subdivision parameter and τ, the time step. Third, the global superconvergent results are obtained by interpolated postprocessing technique. Finally, some numerical results are carried out to confirm the theoretical analysis.     

Superconvergence analysis of a nonconforming finite element method for monotone semilinear elliptic optimal control problems

A nonconforming finite element method (FEM) is proposed for optimal control problems (OCPs) governed by monotone semilinear elliptic equations. The state and adjoint state are approximated by the nonconforming elements, and the control is approximated by the orthogonal projection of the adjoint state, respectively. Some global supercloseness and superconvergence estimates are achieved by making full use of the distinguish characters of this element, such as the interpolation equals to its Ritz projection, and the consistency error is 1 − ε ( is small enough) order higher than its interpolation error in the broken energy norm when the exact solution belongs to H^{3 − ε}(Ω). Finally, some numerical results are presented to verify the theoretical analysis.     

Adaptive stabilized mixed finite volume methods for the incompressible flow

In this article, we study adaptive stabilized mixed finite volume methods for the incompressible flows approximated using the lower order elements. A residual type of a posteriori error estimator is designed and studied with the derivation of upper and lower bounds between the exact solution and the finite volume solution. A discrete local lower bound between two successive finite volume solutions is also obtained. Also, convergence of the adaptive stabilized mixed finite volume methods is established. The presented methods have three prominent features. First, it is of practical convenience in real applications with the same partitions for velocity and pressure. Second, less computational time is required by easily applying both the lower order elements and the local grid refinement necessary for the elements of interest. Third, compared with the standard finite element method, its analysis of H¹‐norm and L²‐norm for the velocity and pressure are usually derived without any high order regularity conditions on the exact solution. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 31: 1424–1443, 2015     

Galerkin discontinuous approximation of the transport equation and viscoelastic fluid flow on quadrilaterals

Numerical simulation of industrial processes involving viscoelastic liquids is often based on finite element methods on quadrilateral meshes. However, numerical analysis of these methods has so far been limited to triangular meshes. In this work, we consider quadrilateral meshes. We first study the approximation of the transport equation by a Galerkin discontinuous method and prove an 𝒪(h^k+1/2) error estimates for the Q_k finite element. Then we study a differential model for viscoelastic flow with unknowns u the velocity, p the pressure, and σ the viscoelastic part of the extra-stress tensor. The approximations are ((Q₁)² transforms of) Q_k+1 continuous for u, Q_k discontinuous for σ, and P_k discontinuous for p, with k ≥ 1. Upwinding for σ is obtained by the Galerkin discontinuous method. We show that an error estimate of order 𝒪(h^k+1/2) is valid in the energy norm for the three unknowns. © 1998 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 14: 97–114, 1998     

Two lower order nonconforming quadrilateral elements for the Reissner-Mindlin plate

This paper generalizes two nonconforming rectangular elements of the Reissner-Mindlin plate to the quadrilateral mesh. The first quadrilateral element uses the usual conforming bilinear element to approximate both components of the rotation, and the modified nonconforming rotated Q ₁ element enriched with the intersected term on each element to approximate the displacement, whereas the second one uses the enriched modified nonconforming rotated Q ₁ element to approximate both the rotation and the displacement. Both elements employ a more complicated shear force space to overcome the shear force locking, which will be described in detail in the introduction. We prove that both methods converge at optimal rates uniformly in the plate thickness t and the mesh distortion parameter in both the H ¹-and the L ²-norms, and consequently they are locking free. This work was supported by the National Natural Science Foundation of China (Grant No. 10601003) and National Excellent Doctoral Dissertation of China (Grant No. 200718)     

A mixed nonconforming finite element for linear elasticity

This article considers a mixed finite element method for linear elasticity. It is based on a modified mixed formulation that enforces the continuity of the stress weakly by adding a jump term of the approximated stress on interior edges. The symmetric stress are approximated by nonconforming linear elements and the displacement by piecewise constants. We establish ??(h) error bound in the (broken) L² norm for the divergence of the stress and ??(h) error bound in the L² norm for both the displacement and the stress tensor. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005.     

An optimal order error estimate for an upwind discretization of the Navier—Stokes equations

We analyze a finite-element approximation of the stationary incompressible Navier–Stokes equations in primitive variables. This approximation is based on the nonconforming P₁/P₀ element pair of Crouzeix/Raviart and a special upwind discretization of the convective term. An optimal error estimate in a discrete H¹-norm for the velocity and in the L²-norm for the pressure is proved. Some numerical results are presented. © 1996 John Wiley & Sons, Inc.     

Uniformly convergent C 0‐nonconforming triangular prism element for fourth‐order elliptic singular perturbation problem

In this article, we introduce a C ⁰‐nonconforming triangular prism element for the fourth‐order elliptic singular perturbation problem in three dimensions by using the bubble functions. The element is proved to be convergent in the energy norm uniformly with respect to the perturbation parameter. © 2014 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 1785–1796, 2014     

Nonconforming spline collocation methods in irregular domains

This article studies a class of nonconforming spline collocation methods for solving elliptic PDEs in an irregular region with either triangular or quadrilateral partition. In the methods, classical Gaussian points are used as matching points and the special quadrature points in a triangle or quadrilateral element are used as collocation points. The solution and its normal derivative are imposed to be continuous at the marching points. The authors present theoretically the existence and uniqueness of the numerical solution as well as the optimal error estimate in H¹‐norm for a spline collocation method with rectangular elements. Numerical results confirm the theoretical analysis and illustrate the high‐order accuracy and some superconvergence features of methods. Finally the authors apply the methods for solving two physical problems in compressible flow and linear elasticity, respectively. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Stabilized mixed finite element methods for the Navier‐Stokes equations with damping

In this paper, the stabilized mixed finite element methods are presented for the Navier‐Stokes equations with damping. The existence and uniqueness of the weak solutions are proven by use of the Brouwer fixed‐point theorem. Then, optimal error estimates for the H¹‐norm and L²‐norm of the velocity and the L²‐norm of the pressure are derived. Moreover, on the basis of the optimal L²‐norm error estimate of the velocity, a stabilized two‐step method is proposed, which is more efficient than the usual stabilized methods. Finally, two numerical examples are implemented to confirm the theoretical analysis.     

Optimal estimates on stabilized finite volume methods for the incompressible Navier–Stokes model in three dimensions

Optimal estimates on stabilized finite volume methods for the three dimensional Navier–Stokes model are investigated and developed in this paper. Based on the global existence theorem [23], we first prove the global bound for the velocity in the H¹‐norm in time of a solution for suitably small data, and uniqueness of a suitably small solution by contradiction. Then, a full set of estimates is then obtained by some classical Galerkin techniques based on the relationship between finite element methods and finite volume methods approximated by the lower order finite elements for the three dimensional Navier–Stokes model.     

On stability of the P n mod /P n element for incompressible flow problems

It is well known that finite element spaces used for approximating the velocity and the pressure in an incompressible flow problem have to be stable in the sense of the inf-sup condition of Babuška and Brezzi if a stabilization of the incompressibility constraint is not applied. In this paper we consider a recently introduced class of triangular nonconforming finite elements of nth order accuracy in the energy norm called P _n ^mod elements. For n ≤ 3 we show that the stability condition holds if the velocity space is constructed using the P _n ^mod elements and the pressure space consists of continuous piecewise polynomial functions of degree n. This research has been supported by the Grant Agency of the Czech Republic under the grant No. 201/05/0005 and by the grant MSM 0021620839.