Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators

We discuss several adaptive mesh-refinement strategies based on (h − h/2)-error estimation. This class of adaptive methods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead. We prove that, under the saturation assumption, these adaptive algorithms are convergent. Our framework applies not only to finite element methods, but also yields a first convergence proof for adaptive boundary element schemes. For a finite element model problem, we extend the proposed adaptive scheme and prove convergence even if the saturation assumption fails to hold in general.     

Iterative schemes for mixed finite element methods with applications to elasticity and compressible flow problems

Summary Iterative schemes for mixed finite element methods are proposed and analyzed in two abstract formulations. The first one has applications to elliptic equations and incompressible fluid flow problems, while the second has applications to linear elasticity and compressible Stokes problems. These schemes are constructed through iteratively penalizing the mixed finite element scheme, of which iterated penalty method and augmented Lagrangian method are special cases. Convergence theorems are demonstrated in abstract formulations in Hilbert spaces, and applications to individual physical problems are considered as examples. Theoretical analysis and computational experiments both show that the proposed schemes have very fast convergence; a few iterations are normally enough to reduce the iterative error to a prescribed precision. Numerical examples with continuous and discontinuous coefficients are presented.     

A general mixed covolume framework for constructing conservative schemes for elliptic problems

We present a general framework for the finite volume or covolume schemes developed for second order elliptic problems in mixed form, i.e., written as first order systems. We connect these schemes to standard mixed finite element methods via a one-to-one transfer operator between trial and test spaces. In the nonsymmetric case (convection-diffusion equation) we show one-half order convergence rate for the flux variable which is approximated either by the lowest order Raviart-Thomas space or by its image in the space of discontinuous piecewise constants. In the symmetric case (diffusion equation) a first order convergence rate is obtained for both the state variable (e.g., concentration) and its flux. Numerical experiments are included.

     

一类非线性双曲-抛物耦合问题的A.D.I.Galerkin方法

1　引　言考虑三维非线性双曲 -抛物耦合初边值问题 :utt- . (a1 (X,t,u) u) +b1 (X,t,u,v) . u　　　　 +α1 e. v =f(X,t,u,v) ,X∈Ω,t∈ J.vt-a2 Δv +b2 (X,t,u,v) . v　　　　 +α2 e. ut=g(X,t,u,v) ,X∈Ω,t∈ J.u(X,t) =v(X,t) =0 , X∈ Ω ,t∈ J.u(X,0 ) =u0 (X) ,ut(X,0 ) =ut0 (X) ,v(X,0 ) =v0 (X) ,X∈Ω.(1 .1 )其中 ,X=(x1 ,x2 ,x3) ,Ω=(c1 ,d1 )× (c2 ,d2 )× (c3,d3)为 R3中矩形区域 ,边界 Ω . J=[0 ,T] ,T>0为一正常数 .b1 ,b2 ,f,g均为已知光滑函数 (其中 b1 ,b2 为向量函数 ) ,且关于 u,v满足 L…     

MIXED FINITE ELEMENT METHODS FOR FRACTIONAL NAVIER-STOKES EQUATIONS

This paper gives the detailed numerical analysis of mixed finite element method for fractional Navier-Stokes equations.The proposed method is based on the mixed finite element method in space and a finite difference scheme in time.The stability analyses of semi-discretization scheme and fully discrete scheme are discussed in detail.Furthermore,We give the convergence analysis for both semidiscrete and flly discrete schemes and then prove that the numerical solution converges the exact one with order O(h2+k),where h and k:respectively denote the space step size and the time step size.Finally,numerical examples are presented to demonstrate the effectiveness of our numerical methods.     

Primal mixed formulations for the coupling of FEM and BEM. Part I: linear problems

We apply the boundary integral equation method and a primal mixed finite element approach to study the weak solvability and Galerkin approximations of linear interior transmission problems arising in potential theory and elastostatics. The existence and uniqueness of solution of the resulting weak formulations and of the associated discrete schemes are derived by using the classical theory for variational problems with constraints. Suitable finite element subspaces of Lagrange type satisfying the compatibility conditions are utilized for defining the Galerkin scheme. The error analysis and corresponding rates of convergence are also provided.     

An Iterative Tikhonov Regularization for Solving Singularly Perturbed Elliptic PDE

In this paper, we consider a class of singularly perturbed elliptical problems with homogeneous boundary conditions. We consider a regularized iterative method for solving such problems. Convergence analysis and error estimate are derived. The regularization parameter is chosen according to an a priori strategy. We give numerical results to illustrate that the method is implementable compared with numerical methods such as Shishkin and finite element schemes. The study demonstrates that the iterated regularized scheme can be considered as an alternate method for solving singularly perturbed elliptical problems.     

复Schrdinger场和实Boussinesq场耦合作用下一类方程组的有限元分析

复Schrodinger场和实Boussinesq场相互作用下的孤立子问题,已经引起广泛的研究.这类问题孤立子解的最重要特征在于孤立子之间的相互作用是非弹性的,见[1].在[1]中讨论了复Schrodinger场和实Boussinesq场耦合作用下一类方程组:     

Convergence of Extrapolated BDF2 Finite Element Schemes for Unsteady Penetrative Convection Model

Extrapolated two-step backward difference (BDF2) in time and finite element in space discretization for the unsteady penetrative convection model is analyzed. Penetrative convection model employs a nonlinear equation of state making the problem more nonlinear. Optimal order error estimates are derived for the semi-discrete finite element spatial discretization. Two time discretization schemes based on linear extrapolation are proposed and analyzed, namely a coupled and a decoupled scheme. In particular, we show that although both schemes are unconditionally nonlinearly stable, the decoupled scheme converges unconditionally whereas coupled scheme requires that the time step be sufficiently small for convergence. These time discretization schemes can be implemented efficiently in practice, saving computational memory. Numerical computations and numerical convergence checks are presented to demonstrate the efficiency and the accuracy of the schemes.     

Error estimates of H^1-Galerkin mixed finite element method for Schrodinger equation

An H1-Galerkin mixed finite element method is discussed for a class of second order SchrSdinger equation. Optimal error estimates of semidiscrete schemes are derived for problems in one space dimension. At the same time, optimal error estimates are derived for fully discrete schemes. And it is showed that the H1-Galerkin mixed finite element approximations have the same rate of convergence as in the classical mixed finite element methods without requiring the LBB consistency condition.     

Approximation of weak adjoints by reverse automatic differentiation of BDF methods

We shed light on the relation between the discrete adjoints of multistep backward differentiation formula (BDF) methods and the solution of the adjoint differential equation. To this end, we develop a functional-analytic framework based on a constrained variational problem and introduce the notion of weak adjoint solutions of ordinary differential equations. We devise a Petrov-Galerkin finite element (FE) interpretation of the BDF method and its discrete adjoint scheme obtained by reverse internal numerical differentiation. We show how the FE approximation of the weak adjoint is computed by the discrete adjoint scheme and prove its convergence in the space of normalized functions of bounded variation. We also show convergence of the discrete adjoints to the classical adjoints on the inner time interval. Finally, we give numerical results for non-adaptive and fully adaptive BDF schemes. The presented framework opens the way to carry over techniques on global error estimation from FE methods to BDF methods.     

-curved finite elements and applications to plate and shell problems

Many thin-plate and thin-shell problems are set on plane reference domains with a curved boundary. Their approximation by conforming finite-elements methods requires ¹-curved finite elements entirely compatible with the associated ¹-rectilinear finite elements. In this contribution we introduce a ¹-curved finite element compatible with the P₅-Argyris element, we study its approximation properties, and then, we use such an element to approximate the solution of thin-plate or thin-shell problems set on a plane-curved boundary domain. We prove the convergence and we get a priori asymptotic error estimates which show the very high degree of accuracy of the method. Moreover we obtain criteria to observe when choosing the numerical integration schemes in order to preserve the order of the error estimates obtained for exact integration.     

一类退化非线性抛物型方程组的变网格有限元方法

本文研究一类退化非线性抛物型方程组的变网格有限元方法.我们从油、水两相渗流驱动问题的实际需要出发,研究在求解过程中对不同的时刻空间区域采用不同的有限元网格.例如在两相驱动问题中,油、水前沿的位置将随时间而向前推移,从而前沿曲     

Finite volume element approximation and analysis for a kind of semiconductor device simulation

In this paper, we present a finite volume element scheme for a kind of two dimensional semiconductor device simulation. A general framework is developed for finite volume element approximation of the semiconductor problems. We construct a fully discrete finite volume element scheme based on triangulations with a piecewise linear finite element space and a general type of control volume. Optimal-order convergence in H ¹-norm is derived.     

PPIFE Method with Non-Homogeneous Flux Jump Conditions and Its Efficient Numerical Solver for Elliptic Optimal Control Problems with Interfaces

In this paper, we design a partially penalized immersed finite element method for solving elliptic interface problems with non-homogeneous flux jump conditions. The method presented here has the same global degrees of freedom as classic immersed finite element method. The non-homogeneous flux jump conditions can be handled accurately by additional immersed finite element functions. Four numerical examples are provided to demonstrate the optimal convergence rates of the method in $L^{\infty}$, $L^{2}$ and $H^{1}$ norms. Furthermore, the method is combined with post-processing technique to solve elliptic optimal control problems with interfaces. To solve the resulting large-scale system, block diagonal preconditioners are introduced. These preconditioners can lead to fast convergence of the Krylov subspace methods such as GMRES and are independent of the mesh size. Four numerical examples are presented to illustrate the efficiency of the numerical schemes and preconditioners.     

One- and two-level Schwarz methods for variational inequalities of the second kind and their application to frictional contact

We present and analyze subspace correction methods for the solution of variational inequalities of the second kind and apply these theoretical results to non smooth contact problems in linear elasticity with Tresca and non-local Coulomb friction. We introduce these methods in a reflexive Banach space, prove that they are globally convergent and give error estimates. In the context of finite element discretizations, where our methods turn out to be one- and two-level Schwarz methods, we specify their convergence rate and its dependence on the discretization parameters and conclude that our methods converge optimally. Transferring this results to frictional contact problems, we thus can overcome the mesh dependence of some fixed-point schemes which are commonly employed for contact problems with Coulomb friction.     

On the Construction and Analysis of Finite Volume Element Schemes with Optimal $L^2$ Convergence Rate

We provide a general construction method for a finite volume element (FVE) scheme with the optimal $L^2$ convergence rate. The $k$-($k$-1)-order orthogonal condition (generalized) is proved to be a sufficient and necessary condition for a $k$-order FVE scheme to have the optimal $L^2$ convergence rate in 1D, in which the independent dual parameters constitute a ($k$-1)-dimension surface in the reasonable domain in $k$-dimension.In the analysis, the dual strategies in different primary elements are not necessarily to be the same, and they are allowed to be asymmetric in each primary element, which open up more possibilities of the FVE schemes to be applied to some complex problems, such as the convection-dominated problems. It worth mentioning that, the construction can be extended to the quadrilateral meshes in 2D. The stability and $H^1$ estimate are proved for completeness. All the above results are demonstrated by numerical experiments.     

美式期权定价问题的数值方法

本文研究美式股票看跌期权定价问题的数值方法。通过将问题转化为等价的变分不等式方程，分别建立了半离散和全离散有限元逼近格式。并给出了有限元解的收敛性和稳定性分析。数值实验表明本文算法是一个高效和收敛的算法。     

High-order linearly implicit two-step peer – finite element methods for time-dependent PDEs

Linearly-implicit two-step peer methods are successfully applied in the numerical solution of ordinary differential and differential-algebraic equations. One of their strengths is that even high-order methods do not show order reduction in computations for stiff problems. With this property, peer methods commend themselves as time-stepping schemes in finite element calculations for time-dependent partial differential equations (PDEs).We have included a class of linearly-implicit two-step peer methods in the finite element software Kardos. There PDEs are solved following the Rothe method, i.e. first discretised in time, leading to linear elliptic problems in each stage of the peer method. We describe the construction of the methods and how they fit into the finite element framework. We also discuss the starting procedure of the two-step scheme and questions of local temporal error control.The implementation is tested for two-step peer methods of orders three to five on a selection of PDE test problems on fixed spatial grids. No order reduction is observed and the two-step methods are more efficient, at least competitive, in comparison with the linearly implicit one-step methods provided in Kardos.