Finite element approximation of a fourth order nonlinear degenerate parabolic equation

Summary. We consider a fully practical finite element approximation of the fourth order nonlinear degenerate parabolic equation where generically for any given . An iterative scheme for solving the resulting nonlinear discrete system is analysed. In addition to showing well-posedness of our approximation, we prove convergence in one space dimension. Finally some numerical experiments are presented. Received July 29, 1997     

Analysis of a pressure-stabilized finite element approximation of the stationary Navier-Stokes equations

Summary. The purpose of this paper is to analyze a finite element approximation of the stationary Navier-Stokes equations that allows the use of equal velocity-pressure interpolation. The idea is to introduce as unknown of the discrete problem the projection of the pressure gradient (multiplied by suitable algorithmic parameters) onto the space of continuous vector fields. The difference between these two vectors (pressure gradient and projection) is introduced in the continuity equation. The resulting formulation is shown to be stable and optimally convergent, both in a norm associated to the problem and in the norm for both velocities and pressure. This is proved first for the Stokes problem, and then it is extended to the nonlinear case. All the analysis relies on an inf-sup condition that is much weaker than for the standard Galerkin approximation, in spite of the fact that the present method is only a minor modification of this. Received May 25, 1998 / Revised version received August 31, 1999 / Published online July 12, 2000     

Convergence of a finite element scheme for the two-dimensional time-dependent Schrödinger equation in a long strip

This paper addresses the finite element method for the two-dimensional time-dependent Schrödinger equation on an infinite strip by using artificial boundary conditions. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying the Crank-Nicolson scheme in time and a bilinear or quadratic finite element approximation in space. This scheme, by a rigorous analysis, has been proved to be unconditionally stable and convergent, and its convergence order has also been obtained. Finally, two numerical examples are given to verify the accuracy of the scheme.     

A defect-correction method for unsteady conduction-convection problems II: Time discretization

In this paper, a fully discrete defect-correction mixed finite element method (MFEM) for solving the non-stationary conduction-convection problems in two dimension, which is leaded by combining the Back Euler time discretization with the two-step defect correction in space, is presented. In this method, we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct these solutions on the same grid using a linearized defect-correction technique. The stability and the error analysis are derived. The theory analysis shows that our method is stable and has a good convergence property. Some numerical results are also given, which show that this method is highly efficient for the unsteady conduction-convection problems.     

Lagrange multiplier method with penalty for elliptic and parabolic interface problems

The basic requirement for the stability of the mortar element method is to construct finite element spaces which satisfy certain criteria known as inf-sup (well known as LBB, i.e., Ladyzhenskaya-Babu?ka-Brezzi) condition. Many natural and convenient choices of finite element spaces are ruled out as these spaces may not satisfy the inf-sup condition. In order to alleviate this problem Lagrange multiplier method with penalty is used in this paper. The existence and uniqueness results of the discrete problem are discussed without using the discrete LBB condition. We have also analyzed the Lagrange multiplier method with penalty for parabolic initial-boundary value problems using semidiscrete and fully discrete schemes. We have derived sub-optimal order of estimates for both semidiscrete and fully discrete schemes. The results of numerical experiments support the theoretical results obtained in this article.     

A priori and a posteriori error analyses in the study of viscoelastic problems

In this work, the numerical approximation of a viscoelastic problem is studied. A fully discrete scheme is introduced by using the finite element method to approximate the spatial variable and an Euler scheme to discretize time derivatives. Then, two numerical analyses are presented. First, a priori estimates are proved from which the linear convergence of the algorithm is derived under suitable regularity conditions. Secondly, an a posteriori error analysis is provided extending some preliminary results obtained in the study of the heat equation. Upper and lower error bounds are obtained.     

Numerical analysis of the Cahn-Hilliard equation with a logarithmic free energy

Summary A fully discrete finite element method for the Cahn-Hilliard equation with a logarithmic free energy based on the backward Euler method is analysed. Existence and uniqueness of the numerical solution and its convergence to the solution of the continuous problem are proved. Two iterative schemes to solve the resulting algebraic problem are proposed and some numerical results in one space dimension are presented.     

半线性抛物最优控制问题全离散插值系数有限元方法的收敛性分析

本文考虑全离散插值系数有限元方法求解半线性抛物最优控制问题,其中控制变量用分片常数函数逼近,状态变量和对偶状态变量用分片线性函数逼近.对于方程中的半线性项,先用插值系数技巧处理,再用牛顿迭代法求解.通过引入一些辅助变量和投影算子,并利用有限元空间的逼近性质,得到半线性抛物最优控制问题插值系数有限元方法的收敛性结果;数值算例结果验证了理论结果的正确性.     

A fully discrete stabilized finite element method for the time-dependent Navier-Stokes equations

In this article, we consider a fully discrete stabilized finite element method based on two local Gauss integrations for the two-dimensional time-dependent Navier-Stokes equations. It focuses on the lowest equal-order velocity-pressure pairs. Unlike the other stabilized method, the present approach does not require specification of a stabilization parameter or calculation of higher-order derivatives, and always leads to a symmetric linear system. The Euler semi-implicit scheme is used for the time discretization. It is shown that the proposed fully discrete stabilized finite element method results in the optimal order bounds for the velocity and pressure.     

A stabilized Lagrange multiplier method for the finite element approximation of contact problems in elastostatics

In this work we consider a stabilized Lagrange (or Kuhn–Tucker) multiplier method in order to approximate the unilateral contact model in linear elastostatics. The particularity of the method is that no discrete inf-sup condition is needed in the convergence analysis. We propose three approximations of the contact conditions well adapted to this method and we study the convergence of the discrete solutions. Several numerical examples in two and three space dimensions illustrate the theoretical results and show the capabilities of the method.     

LEGENDRE SPECTRAL-FINITE ELEMENT METHOD FOR THE THREE-DIMENSIONAL UNSTEADY NAVIER-STOKES EQUATION

This paper is devoted to the mixed Legendre spectral-finite element approximation of the three-dimensional, non-periodic, unsteady Navier-Stokes equations. A class of fully discrete schemes are constructed with artificial compression. The generalized stability and convergence are proved strictly on the assumption that the two-dimensional inf-sup condition of the finite element approximation is satisfied.     

A characteristics-mixed covolume method for a convection-dominated transport problem

In this paper, we propose a characteristics-mixed covolume method for approximating the solution to a convection dominated transport problem. The method is a combination of characteristic approximation to handle the convection term in time and mixed covolume method spatial approximation to deal with the diffusion term. The velocity and press are approximated by the lowest order Raviart-Thomas mixed finite element space on rectangles. The projection of a mixed covolume element is introduced. We prove its first order optimal rate of convergence for the approximate velocities in the L² norm as well as for the approximate pressures in the L² norm.     

Finite element method for viscoelastic fluids flows: approximation of the Maxwell model

We discuss about the finite element approximation of viscoelastic fluid flow. We consider a fluid obeying the Oldroyd model and particularly we study the purely viscoelastic case, the so-called Maxwell model, important in practice for the applications. We examine two kinds of methods used for the approximation of the Maxwell model: method using a splitting technique and finite element method satisfying inf-sup conditions relating tensor and velocity. We present numerical results for these methods and we discuss about their stability. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A combined finite volume–nonconforming/mixed-hybrid finite element scheme for degenerate parabolic problems

We propose and analyze a numerical scheme for nonlinear degenerate parabolic convection–diffusion–reaction equations in two or three space dimensions. We discretize the diffusion term, which generally involves an inhomogeneous and anisotropic diffusion tensor, over an unstructured simplicial mesh of the space domain by means of the piecewise linear nonconforming (Crouzeix–Raviart) finite element method, or using the stiffness matrix of the hybridization of the lowest-order Raviart–Thomas mixed finite element method. The other terms are discretized by means of a cell-centered finite volume scheme on a dual mesh, where the dual volumes are constructed around the sides of the original mesh. Checking the local Péclet number, we set up the exact necessary amount of upstream weighting to avoid spurious oscillations in the convection-dominated case. This technique also ensures the validity of the discrete maximum principle under some conditions on the mesh and the diffusion tensor. We prove the convergence of the scheme, only supposing the shape regularity condition for the original mesh. We use a priori estimates and the Kolmogorov relative compactness theorem for this purpose. The proposed scheme is robust, only 5-point (7-point in space dimension three), locally conservative, efficient, and stable, which is confirmed by numerical experiments.This work was supported by the GdR MoMaS, CNRS-2439, ANDRA, BRGM, CEA, EdF, France.     

Error estimate of fully discrete discontinuous Galerkin approximation for the viscoelasticity type equation

In this paper we develop fully discrete discontinuous Galerkin approximation using symmetric interior penalty method. We construct finite element spaces which consist of piecewise continuous polynomials. We introduce an appropriate elliptic-type projection of u and prove its optimal convergence. We develop fully discrete discontinuous Galerkin approximations and prove the optimal convergence in ? ^∞(L ²) normed space.     

A TWO-LEVEL FINITE ELEMENT GALERKIN METHOD FOR THE NONSTATIONARY NAVIER-STOKES EQUATIONS II： TIME DISCRETIZATION

In this article we consider the fully discrete two-level finite element Galerkin method for the two-dimensional nonstationary incompressible Navier-Stokes equations. This method consists in dealing with the fully discrete nonlinear Navier-Stokes problem on a coarse mesh with width $H$ and the fully discrete linear generalized Stokes problem on a fine mesh with width $h << H$. Our results show that if we choose $H=O(h^{1/2}$) this method is as the same stability and convergence as the fully discrete standard finite element Galerkin method which needs dealing with the fully discrete nonlinear Navier-Stokes problem on a fine mesh with width $h$. However, our method is cheaper than the standard fully discrete finite element Galerkin method.     

Nonoverlapping Domain Decomposition Method with Mixed Element for Elliptic Problems

1.IntroductionDomaindecompositionasanewmethodofcomputationalmathematics,waJsdevel-opedsincethedevelopmentofparallelcomputersandmultiprocessorsupercomputers-Usingdomaindecompositionwecandecreasethescaleoftheproblemandimplementthesub-problemsonparallelcomputer.Fromatechnicalpointofviewmostofdo-maindecompositionmethodsconsideredsofarhavebeendealingwithfiniteelementmethods.In[1,2]ZhangandHuanghavegivenakindofnonoverlappingdomaindecompositionprocedurewithpiecewiselinearfiniteelementapproximation.…     

Finite element approximation of spatially extended predator–prey interactions with the Holling type II functional response

We study the numerical approximation of the solutions of a class of nonlinear reaction–diffusion systems modelling predator–prey interactions, where the local growth of prey is logistic and the predator displays the Holling type II functional response. The fully discrete scheme results from a finite element discretisation in space (with lumped mass) and a semi-implicit discretisation in time. We establish a priori estimates and error bounds for the semi discrete and fully discrete finite element approximations. Numerical results illustrating the theoretical results and spatiotemporal phenomena are presented in one and two space dimensions. The class of problems studied in this paper are real experimental systems where the parameters are associated with real kinetics, expressed in nondimensional form. The theoretical techniques were adapted from a previous study of an idealised reaction–diffusion system (Garvie and Blowey in Eur J Appl Math 16(5):621–646, 2005).     

Error estimates for a mixed finite element discretization of some degenerate parabolic equations

We consider a numerical scheme for a class of degenerate parabolic equations, including both slow and fast diffusion cases. A particular example in this sense is the Richards equation modeling the flow in porous media. The numerical scheme is based on the mixed finite element method (MFEM) in space, and is of one step implicit in time. The lowest order Raviart–Thomas elements are used. Here we extend the results in Radu et al. (SIAM J Numer Anal 42:1452–1478, 2004), Schneid et al. (Numer Math 98:353–370, 2004) to a more general framework, by allowing for both types of degeneracies. We derive error estimates in terms of the discretization parameters and show the convergence of the scheme. The features of the MFEM, especially of the lowest order Raviart–Thomas elements, are now fully exploited in the proof of the convergence. The paper is concluded by numerical examples.     

Higher order unfitted FEM for Stokes interface problems

We consider the discretization of a stationary Stokes interface problem in a velocity-pressure formulation. The interface is described implicitly as the zero level of a scalar function as it is common in level set based methods. Hence, the interface is not aligned with the mesh. An unfitted finite element discretization based on a Taylor-Hood velocity-pressure pair and an XFEM (or CutFEM) modification is used for the approximation of the solution. This allows for the accurate approximation of solutions which have strong or weak discontinuities across interfaces which are not aligned with the mesh. To arrive at a consistent, stable and accurate formulation we require several additional techniques. First, a Nitsche-type formulation is used to implement interface conditions in a weak sense. Secondly, we use the ghost penalty stabilization to obtain an inf-sup stable variational formulation. Finally, for the highly accurate approximation of the implicitly described geometry, we use a combination of a piecewise linear interface reconstruction and a parametric mapping of the underlying mesh. We introduce the method and discuss results of numerical examples. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)