A multiscale finite element method for the solution of transport equations

A multiscale Galerkin finite element scheme based on the residual free bubble function method is proposed to generate stable and accurate solutions for the transport equations namely diffusion-reaction (DR), convection-diffusion (CD) and convection-diffusion-reaction (CDR) equations. These equations show multiscale behavior in reaction or convection dominated situations. The idea is based on the approximation of the definite integral of the interpolation function within the element, instead of the function approximation. The numerical experiments are performed using the bilinear Lagrangian elements. To validate the approach, the numerical results obtained for a benchmark problem are compared with the analytical solution in a wide range of Peclet and Damköhler numbers. The results show that the developed method is capable of generating stable and accurate solutions.     

Convergence of a finite element/ALE method for the Stokes equations in a domain depending on time

We consider the approximation of the unsteady Stokes equations in a time dependent domain when the motion of the domain is given. More precisely, we apply the finite element method to an Arbitrary Lagrangian Eulerian (ALE) formulation of the system. Our main results state the convergence of the solutions of the semi-discretized (with respect to the space variable) and of the fully-discrete problems towards the solutions of the Stokes system.     

Discrete mass conservation for porous media saturated flow

Global and local mass conservation for velocity fields associated with saturated porous media flow have long been recognized as integral components of any numerical scheme attempting to simulate these flows. In this work, we study finite element discretizations for saturated porous media flow that use Taylor–Hood (TH) and Scott–Vogelius (SV) finite elements. The governing equations are modified to include a stabilization term when using the TH elements, and we provide a theoretical result that shows convergence (with respect to the stabilization parameter) to pointwise mass‐conservative solutions. We also provide results using the SV approximation pair. These elements are pointwise divergence free, leading to optimal convergence rates and numerical solutions. We give numerical results to verify our theory and a comparison with standard mixed methods for saturated flow problems. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 625–640, 2014     

一类带有铁磁材料参数的非线性涡流问题的A-φ有限元法

针对带有铁磁材料的非线性涡流问题,其非线性性通常体现在磁场强度和磁感应强度的关系上.本文提出了一种全离散的有限元A-φ格式,分别在时间和空间上采用向后欧拉公式以及节点有限元进行离散.首先,在合适的函数空间里给出时间上的半离散格式,通过考察其弱形式建立相应的适定性理论,并证明近似解收敛于弱解.其次,给出全离散格式并讨论其误差估计.最后,给出两个数值算例以验证理论结果.     

Superconvergence analysis of fully discrete finite element methods for semilinear parabolic optimal control problems

We study the superconvergence property of fully discrete finite element approximation for quadratic optimal control problems governed by semilinear parabolic equations with control constraints. The time discretization is based on difference methods, whereas the space discretization is done using finite element methods. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. First, we define a fully discrete finite element approximation scheme for the semilinear parabolic control problem. Second, we derive the superconvergence properties for the control, the state and the adjoint state. Finally, we do some numerical experiments for illustrating our theoretical results.     

Convergence and nonlinear stability of the Lagrange-Galerkin method for the Navier-Stokes equations

Summary The Lagrange-Galerkin method is a numerical technique for solving convection — dominated diffusion problems, based on combining a special discretisation of the Lagrangian material derivative along particle trajectories with a Galerkin finite element method. We present optimal error estimates for the Lagrange-Galerkin mixed finite element approximation of the Navier-Stokes equations in a velocity/pressure formulation. The method is shown to be nonlinearly stable.     

Optimal Error Estimate of the Penalty Finite Element Method for the Micropolar Fluid Equations

We present an optimal error estimate of the numerical velocity, pressure, and angular velocity for the fully discrete penalty finite element method of the micropolar equations when the parameters ?, Δ t, and h are sufficiently small. In order to obtain this estimate, we present the time discretization of the penalty micropolar equation that is based on the backward Euler scheme; the spatial discretization of the time discretized penalty micropolar equation is based on a finite elements space pair (X _h , M _h ) that satisfies some approximations properties.     

Approximation dans un Problème de Viscoélasticité Périodique

We are concerned with equations which derive from a quasistatic periodic problem of viscoelasticity. We give a condition which yields to existence and uniqueness of a periodic solution. Then we prove a finite element method based on equilibrium elements for the space approximation and on the explicit Euler scheme for the time approximation.     

Analysis of optimal preconditioners for CutFEM

In this article, we consider a class of unfitted finite element methods for scalar elliptic problems. These so-called CutFEM methods use standard finite element spaces on a fixed unfitted triangulation combined with the Nitsche technique and a ghost penalty stabilization. As a model problem we consider the application of such a method to the Poisson interface problem. We introduce and analyze a new class of preconditioners that is based on a subspace decomposition approach. The unfitted finite element space is split into two subspaces, where one subspace is the standard finite element space associated to the background mesh and the second subspace is spanned by all cut basis functions corresponding to nodes on the cut elements. We will show that this splitting is stable, uniformly in the discretization parameter and in the location of the interface in the triangulation. Based on this we introduce an efficient preconditioner that is uniformly spectrally equivalent to the stiffness matrix. Using a similar splitting, it is shown that the same preconditioning approach can also be applied to a fictitious domain CutFEM discretization of the Poisson equation. Results of numerical experiments are included that illustrate optimality of such preconditioners for the Poisson interface problem and the Poisson fictitious domain problem.     

A combined finite volume–finite element scheme for the discretization of strongly nonlinear convection–diffusion–reaction problems on nonmatching grids

We propose and analyze in this paper a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the time evolution, convection, reaction, and source terms on a given grid, which can be nonmatching and can contain nonconvex elements, by means of the cell‐centered finite volume method. To discretize the diffusion term, we construct a conforming simplicial mesh with the vertices given by the original grid and use the conforming piecewise linear finite element method. In this way, the scheme is fully consistent and the discrete solution is naturally continuous across the interfaces between the subdomains with nonmatching grids, without introducing any supplementary equations and unknowns or using any interpolation at the interfaces. We allow for general inhomogeneous and anisotropic diffusion–dispersion tensors, propose two variants corresponding respectively to arithmetic and harmonic averaging, and use the local Péclet upstream weighting in order to only add the minimal numerical diffusion necessary to avoid spurious oscillations in the convection‐dominated case. The scheme is robust, efficient since it leads to positive definite matrices and one unknown per element, locally conservative, and satisfies the discrete maximum principle under the conditions on the simplicial mesh and the diffusion tensor usual in the finite element method. We prove its convergence using a priori estimates and the Kolmogorov relative compactness theorem and illustrate its behavior on a numerical experiment. © 2009 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2010     

Modeling and analysis of the unilateral contact of a piezoelectric body with a conductive support

We consider a mathematical model which describes the quasistatic process of contact between a piezoelectric body and an electrically conductive support, the so-called foundation. We model the material's behavior with a nonlinear electro-viscoelastic constitutive law; the contact is frictionless and is described with the Signorini condition and a regularized electrical conductivity condition. We derive a variational formulation for the problem and then we prove the existence of a unique weak solution to the model. The proof is based on arguments of nonlinear equations with multivalued maximal monotone operators and fixed point. Then we introduce a fully discrete scheme, based on the finite element method to approximate the spatial variable and the backward Euler scheme to discretize the time derivatives. We treat the unilateral contact conditions by using an augmented Lagrangian approach. We implement this scheme in a numerical code then we present numerical simulations in the study of two-dimensional test problems, together with various comments and interpretations.     

Approximation of the dual problem for error estimation in inelastic problems

In numerical simulations with the finite element method the dependency on the mesh – and for time-dependent problems on the time discretization – arises. Adaptive refinements in space (and time) based on goal-oriented error estimation [1] become more and more popular for finite element analyses to balance computational effort and accuracy of the solution. The introduction of a goal quantity of interest defines a dual problem which has to be solved to estimate the error with respect to it. Often such procedures are based on a space-time Galerkin framework for instationary problems [2]. Discretization results in systems of equations in which the unknowns are nodal values. Contrary, in current finite element implementations for path-dependent problems some quantities storing information about the path-dependence are located at the integration points of the finite elements [3], e.g. plastic strains etc. In this contribution we propose an approach – similar to [4] for sensitivity analysis – for the approximation of the dual problem which mainly maintains the structure of current finite element implementations for path-dependent problems. Here, the dual problem is introduced after discretization. A numerical example illustrates the approach. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Enforcing energy, helicity and strong mass conservation in finite element computations for incompressible Navier-Stokes simulations

We study a finite element scheme for the 3D Navier-Stokes equations (NSE) that globally conserves energy and helicity and, through the use of Scott-Vogelius elements, enforces pointwise the solenoidal constraints for velocity and vorticity. A complete numerical analysis is given, including proofs for conservation laws, unconditional stability and optimal convergence. We also show the method can be efficiently computed by exploiting a connection between this method, its associated penalty method, and the method arising from using grad-div stabilized Taylor-Hood elements. Finally, we give numerical examples which verify the theory and demonstrate the effectiveness of the scheme.     

Penalty finite element approximations for the Stokes equations by L2 projection

This paper considers the penalty finite element method for the Stokes equations, based on some stable finite elements space pair (X_h, M_h) that do satisfy the discrete inf–sup condition. Theoretical results show that the penalty error converges as fast as one should expect from the order of the elements. Moreover, the penalty finite element method by L² projection can improve the penalty error estimates. Finally, we confirm these results by a series of numerical experiments. Copyright © 2008 John Wiley & Sons, Ltd.     

Pure Lagrangian and semi-Lagrangian finite element methods for the numerical solution of Navier–Stokes equations

In this paper we propose a unified formulation to introduce Lagrangian and semi-Lagrangian velocity and displacement methods for solving the Navier–Stokes equations. This formulation allows us to state classical and new numerical methods. Several examples are given. We combine them with finite element methods for spatial discretization. In particular, we propose two new second-order characteristics methods in terms of the displacement, one semi-Lagrangian and the other one pure Lagrangian. The pure Lagrangian displacement methods are useful for solving free surface problems and fluid-structure interaction problems because the computational domain is independent of the time and fluid–solid coupling at the interphase is straightforward. However, for moderate to high-Reynolds number flows, they can lead to high distortion in the mesh elements. When this happens it is necessary to remesh and reinitialize the transformation to the identity. In order to assess the performance of the obtained numerical methods, we solve different problems in two space dimensions. In particular, numerical results for a sloshing problem in a rectangular tank and the flow in a driven cavity are presented.     

A multilevel correction method for Stokes eigenvalue problems and its applications

In this paper, a new multilevel correction scheme is proposed to solve Stokes eigenvalue problems by the finite element method. This new scheme contains a series of correction steps, and the accuracy of eigenpair approximation can be improved after each step. In each correction step, we only need to solve a Stokes problem on the corresponding fine finite element space and a Stokes eigenvalue problem on the coarsest finite element space. This correction scheme can improve the efficiency of solving Stokes eigenvalue problems by the finite element method. As applications of this multilevel correction method, a multigrid method and an adaptive finite element technique are introduced for Stokes eigenvalue problems. Some numerical results are given to validate our schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS GLOBAL FINITE ELEMENT NONLINEAR GALERKIN METHOD FOR THE PENALIZED NAVIER-STOKES EQUATIONS

1. IntroductionIn the numerical simulation of the Navier-Stokes equations one encounters three seriousdifficulties in the case of large Reynolds numbers f the treatment of the incomPressibility con-dition divu = 0, the treatment of the noIilinear terms and the large time integration. For thetreatment of the incoInPressibility condition, one use the penalty method in the case of finiteelemellts [1--2l and for the treatmen of the noulinar terms and the large tfor integration, oneuse the nonlin…     

On the numerical approximations of stiff convection–diffusion equations in a circle

Our aim in this article is to study the numerical solutions of singularly perturbed convection–diffusion problems in a circular domain and provide as well approximation schemes, error estimates and numerical simulations. To resolve the oscillations of classical numerical solutions due to the stiffness of our problem, we construct, via boundary layer analysis, the so-called boundary layer elements which absorb the boundary layer singularities. Using a $P_1$ classical finite element space enriched with the boundary layer elements, we obtain an accurate numerical scheme in a quasi-uniform mesh.     

对流占优的Sobolev方程的投影稳定化有限元方法

对于对流占优的Sobolev方程,提出了一种新的投影稳定化有限元方法,建立了半离散和全离散的投影稳定化格式,给出了解的稳定性和收敛性分析.该方法能够有效克服对流占优,与内罚方法相比,投影格式更简单,计算量更小,且得到的C—N格式是无条件稳定的,时间精度达到了二阶.最后,通过实验证明,数值结果与理论结果完全一致.     

A finite element method for elliptic problems with rapidly oscillating coefficients

In this paper, we consider solving second-order elliptic problems with rapidly oscillating coefficients. Under the assumption that the oscillating coefficients are periodic, on the basis of classical homogenization theory, we present a finite element method whose key is to combine a numerical approximation of the 1-order approximate solution of those equations and a numerical approximation of the classical boundary corrector of those equations from different meshes exploiting the need for different levels of resolution. Numerical experiments are included to illustrate the competitive behavior of the proposed finite element method.