A nonconforming finite element method for a two-dimensional curl–curl and grad-div problem

A numerical method for a two-dimensional curl–curl and grad-div problem is studied in this paper. It is based on a discretization using weakly continuous P ₁ vector fields and includes two consistency terms involving the jumps of the vector fields across element boundaries. Optimal convergence rates (up to an arbitrary positive ) in both the energy norm and the L ₂ norm are established on graded meshes. The theoretical results are confirmed by numerical experiments. The work of the first author was supported in part by the National Science Foundation under Grant No. DMS-03-11790 and by the Humboldt Foundation through her Humboldt Research Award. The work of the third author was supported in part by the National Science Foundation under Grant No. DMS-06-52481.     

The analysis of a finite element method for the Navier–Stokes equations with compressibility

Summary. A finite element formulation is developed for the two dimensional nonlinear time dependent compressible Navier–Stokes equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation is proved. An error estimate for the numerical solution is obtained. Received September 9, 1997 / Revised version received August 12, 1999 / Published online July 12, 2000     

A posteriori error estimates of finite element method for parabolic problems

1IntroductionThebase0fadaPtivecomputing0ffiniteelementmethodisap0steri0rierr0restimates.I.Babuskaisthepioneerinthisfields.Manytechniquesaredevel0pedtoobtainaposteri0rierrorestimators.See[1-3,7-8,19-201.Theyaremainlybased0nthejumps0fthederiva-tivesontheboundariesoftl1eelel11elltandtheresidualintheelemellts.Recelltresultssh0wthatthereareveryclosedrelatiollsbetweellasymptoticexactap0steri0rierrorestimatesandsuperc0nvergence-SeealsoQ.Linetal.[11-13],andChen-Huang['].Therehasbeenmuchprogressill…     

Galerkin finite element method for nonlinear fractional Schrödinger equations

In this paper, a class of nonlinear Riesz space-fractional Schrödinger equations are considered. Based on the standard Galerkin finite element method in space and Crank-Nicolson difference method in time, the semi-discrete and fully discrete systems are constructed. By Brouwer fixed point theorem and fractional Gagliardo-Nirenberg inequality, we prove the fully discrete system is uniquely solvable. Moreover, we focus on a rigorous analysis and consideration of the conservation and convergence properties for the semi-discrete and fully discrete systems. Finally, a linearized iterative finite element algorithm is introduced and some numerical examples are given to confirm the theoretical results.     

The finite volume method based on stabilized finite element for the stationary Navier–Stokes problem

A finite volume method based on stabilized finite element for the two-dimensional stationary Navier–Stokes equations is investigated in this work. A macroelement condition is introduced for constructing the local stabilized formulation for the problem. We obtain the well-posedness of the FVM based on stabilized finite element for the stationary Navier–Stokes equations. Moreover, for quadrilateral and triangular partition, the optimal H¹$H^1$ error estimate of the finite volume solution _uh$u_h$ and L²$L^2$ error estimate for _ph$p_h$ are introduced. Finally, we provide a numerical example to confirm the efficiency of the FVM.     

A mixed finite element method for the unilateral contact problem in elasticity

In this paper, we provide a new mixed finite element approximation of the varia-tional inequality resulting from the unilateral contact problem in elasticity. We use the continuous piecewise P2-P1 finite element to approximate the displacement field and the normal stress component on the contact region. Optimal convergence rates are obtained under the reasonable regularity hypotheses. Numerical example verifies our results.     

Vibration analysis of Euler–Bernoulli nanobeams by using finite element method

In the present study, an efficient finite element model for vibration analysis of a nonlocal Euler–Bernoulli beam has been reported. Nonlocal constitutive equation of Eringen is proposed. Equations of motion for a nonlocal Euler–Bernoulli are derived based on varitional statement. The finite element method is employed to discretize the model and obtain a numerical approximation of the motion equation. The model has been verified with the previously published works and found a good agreement with them. Vibration characteristics, such as fundamental frequencies, are illustrated in graphical and tabulated form. Numerical results are presented to figure out the effects of nonlocal parameter, slenderness ratios, rotator inertia, and boundary conditions on the dynamic characteristics of the beam. The above mention effects play very important role on the dynamic behavior of nanobeams.     

The hp-MITC finite element method for the Reissner–Mindlin plate problem

The popular MITC finite elements used for the approximation of the Reissner–Mindlin plate are extended to the case where elements of non-uniform degree p distribution are used on locally refined meshes. Such an extension is of particular interest to the hp-version and hp-adaptive finite element methods. A priori error bounds are provided showing that the method is locking-free. The analysis is based on new approximation theoretic results for non-uniform Brezzi–Douglas–Fortin–Marini spaces, and extends the results obtained in the case of uniform order approximation on globally quasi-uniform meshes presented by Stenberg and Suri (SIAM J. Numer. Anal. 34 (1997) 544). Numerical examples illustrating the theoretical results and comparing the performance with alternative standard Galerkin approaches are presented for two new benchmark problems with known analytic solution, including the case where the shear stress exhibits a boundary layer. The new method is observed to be locking-free and able to provide exponential rates of convergence even in the presence of boundary layers.     

A quadratic finite element method for solving biharmonic problems in ℝ n

Summary A family of simplicial finite element methods having the simplest possible structure, is introduced to solve biharmonic problems in ⁿ ,n3, using the primal variable. The family is inspired in the MORLEY triangle for the two dimensional case, and in some sense this element can be viewed as its member corresponding to the valuen=2.     

Convergence of a finite element method on a Bakhvalov-type mesh for a singularly perturbed convection–diffusion equation in 2D

A finite element method of any order is applied on a Bakhvalov-type mesh to solve a singularly perturbed convection–diffusion equation in 2D, whose solution exhibits exponential boundary layers. A uniform convergence of (almost) optimal order is proved by means of a carefully defined interpolant.     

An expanded mixed finite element approach via a dual–dual formulation and the minimum residual method

We apply an expanded mixed finite element method, which introduces the gradient as a third explicit unknown, to solve a linear second-order elliptic equation in divergence form. Instead of using the standard dual form, we show that the corresponding variational formulation can be written as a dual–dual operator equation. We establish existence and uniqueness of solution for the continuous and discrete formulations, and provide the corresponding error analysis by using Raviart–Thomas elements. In addition, we show that the corresponding dual–dual linear system can be efficiently solved by a preconditioned minimum residual method. Some numerical results, illustrating this fact and the rate of convergence of the mixed finite element method, are also provided.     

Residual-based a posteriori error estimates of nonconforming finite element method for elliptic problems with Dirac delta source terms

Two residual-based a posteriori error estimators of the nonconforming Crouzeix-Raviart element are derived for elliptic problems with Dirac delta source terms.One estimator is shown to be reliable and efficient,which yields global upper and lower bounds for the error in piecewise W1,p seminorm.The other one is proved to give a global upper bound of the error in Lp-norm.By taking the two estimators as refinement indicators,adaptive algorithms are suggested,which are experimentally shown to attain optimal convergence orders.     

The analysis of a finite element method for the three-species Lotka–Volterra competition-diffusion with Dirichlet boundary conditions

A Galerkin finite element method is developed for the two dimensional/three dimensional nonlinear time-dependent three-species Lotka–Volterra competition-diffusion equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation are proved. An error estimate for the numerical solution is obtained. Numerical computations are carried out to examine the expected orders of accuracy in the error estimates.     

Hybrid coupling of finite element and boundary element methods using Nitsche’s method and the Calderon projection

Numerical Algorithms - In this paper, we discuss a hybridised method for FEM-BEM coupling. The coupling from both sides use a Nitsche-type approach to couple to the trace variable. This leads to a...     

A nonconforming Morley finite element method for the fully nonlinear Monge-Ampère equation

In this paper, we study finite element approximations of the viscosity solution of the fully nonlinear Monge-Ampère equation, det(D ² u) = f (> 0) using the well-known nonconforming Morley element. Our approach is based on the vanishing moment method, which was recently proposed as a constructive way to approximate fully nonlinear second order equations by the author and Feng (J Sci Comput 38(1):74–98, 2009). The vanishing moment method approximates the Monge-Ampère equation by the fourth order quasilinear equation -eD²u^e + det(D²u^e) = f{-\epsilon\Delta^2u^\epsilon + {\rm det}(D^2u^\epsilon) = f} with appropriate boundary conditions. We develop a finite element scheme using the n-dimensional Morley element introduced in Wang and Xu (Numer Math 103:155–169, 2006) to approximate the regularized fourth order problem in two and three dimensions, and then derive optimal order error estimates.     

Two-level defect-correction locally stabilized finite element method for the steady Navier–Stokes equations

This paper proposes a two-level defect-correction stabilized finite element method for the steady Navier–Stokes equations based on local Gauss integration. The method combines the two-level strategy with the defect-correction method under the assumption of the uniqueness condition. Both the simplified and the Newton scheme are proposed and analyzed. Moreover, the numerical illustrations agree completely with the theoretical expectations.     

Some second-order �� schemes combined with finite element method for nonlinear fractional cable equation

Numerical Algorithms - In this article, some second-order time discrete schemes covering parameter ?? combined with Galerkin finite element (FE) method are proposed and analyzed for...