Rational approximation to trigonometric operators

We consider the approximation of trigonometric operator functions that arise in the numerical solution of wave equations by trigonometric integrators. It is well known that Krylov subspace methods for matrix functions without exponential decay show superlinear convergence behavior if the number of steps is larger than the norm of the operator. Thus, Krylov approximations may fail to converge for unbounded operators. In this paper, we propose and analyze a rational Krylov subspace method which converges not only for finite element or finite difference approximations to differential operators but even for abstract, unbounded operators. In contrast to standard Krylov methods, the convergence will be independent of the norm of the operator and thus of its spatial discretization. We will discuss efficient implementations for finite element discretizations and illustrate our analysis with numerical experiments. AMS subject classification (2000)  65F10, 65L60, 65M60, 65N22     

Numerical schemes for Hamilton-Jacobi equations on unstructured meshes

Summary.  In this paper, a numerical scheme is presented by applying the finite element method to the viscosity equations of the Hamilton-Jacobi equations on unstructured meshes. By improving the finite element scheme, another numerical scheme is constructed. Under certain limitations, the numerical solutions of the two schemes converge to the viscosity solutions of the Hamilton-Jacobi equations. The latter numerical scheme has weaker restrictions than the former scheme for convergence. Numerical examples are provided to test the stability, convergence and sensitivity to different meshes. Received November 5, 2001 / Revised version received March 5, 2002 / Published online October 29, 2002 RID="*" ID="*" Current address: Department of Applied Mathematics, University of Petroleum, Dongying 257062, Shandong, P.R.China; e-mail: xianggui_li@sina.com Mathematics Subject Classification (1991): 65M60     

A defect-correction method for unsteady conduction convection problems Ⅰ:spatial discretization

In this paper, a semi-discrete defect-correction mixed finite element method (MFEM) for solving the non-stationary conduction-convection problems in two dimension is presented. In this method, we solve the nonlinear equations with an added artificial viscosity term on a finite element grid and correct this 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 ...     

Numerical analysis of a modified finite element nonlinear Galerkin method

Summary. A fully discrete modified finite element nonlinear Galerkin method is presented for the two-dimensional equation of Navier-Stokes type. The spatial discretization is based on two finite element spaces X_H and X_h defined on a coarse grid with grid size H and a fine grid with grid size h << H, respectively; the time discretization is based on the Euler explicit scheme with respect to the nonlinear term. We analyze the stability and convergence rate of the method. Comparing with the standard finite element Galerkin method and the nonlinear Galerkin method, this method can admit a larger time step under the same convergence rate of same order. Hence this method can save a large amount of computational time. Finally, we provide some numerical tests on this method, the standard finite element Galerkin method, and the nonlinear Galerkin method, which are in a good agreement with the theoretical analysis.Mathematics Subject Classification (2000): 35Q30, 65M60, 65N30, 76D05     

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.     

A semi‐discrete defect correction finite element method for unsteady incompressible magnetohydrodynamics equations

In this report, we give a semi‐discrete defect correction finite element method for the unsteady incompressible magnetohydrodynamics equations. The defect correction method is an iterative improvement technique for increasing the accuracy of a numerical solution without applying a grid refinement. Firstly, the nonlinear magnetohydrodynamics equations is solved with an artificial viscosity term. Then, the numerical solutions are improved on the same grid by a linearized defect‐correction technique. Then, we give the numerical analysis including stability analysis and error analysis. The numerical analysis proves that our method is stable and has an optimal convergence rate. In order to show the effect of our method, some numerical results are shown. Copyright © 2017 John Wiley & Sons, Ltd.     

A two-grid finite element method for the Allen-Cahn equation with the logarithmic potential

In this paper, we present a two-grid finite element method for the Allen-Cahn equation with the logarithmic potential. This method consists of two steps. In the first step, based on a fully implicit finite element method, the Allen-Cahn equation is solved on a coarse grid with mesh size H. In the second step, a linearized system whose nonlinear term is replaced by the value of the first step is solved on a fine grid with mesh size h. We give the energy stabilities of the traditional finite element method and the two-grid finite element method. The optimal convergence order of the two-grid finite element method in H¹ norm is achieved when the mesh sizes satisfy h = O(H²). Numerical examples are given to demonstrate the validity of the proposed scheme. The results show that the two-grid method can save the CPU time while keeping the same convergence rate.     

On finite element methods for convection dominated phenomena

The aim of this work is to give theoretical justification of several types of finite element approximations to the initial-boundary value problems of first order linear hyperbolic equations. Our approximate scheme is obtained by the piecewise linear continuous finite element method for space variable, x, and the Euler type step by step integration method for time variable, t. An artificial viscosity technique, up-stream type methods are considered within the frame work of L²-theory. The convergence and the error estimate of the approximate solutions to the true one are discussed.     

Second order modified method of characteristics mixed defect-correction finite element method for time dependent Navier–Stokes problems

In this paper, a second order modified method of characteristics defect-correction (SOMMOCDC) mixed finite element method for the time dependent Navier–Stokes problems is presented. In this method, the hyperbolic part (the temporal and advection term) are treated by a second order characteristics tracking scheme, and the non-linear term is linearized at the same time. Then, we solve the equations with an added artificial viscosity term and correct this solution by using the defect-correction technique. The error analysis shows that this method has a good convergence property. In order to show the efficiency of the SOMMOCDC mixed finite element method, we first present some numerical results of an analytical solution problem, which agrees very well with our theoretical results. Then, we give some numerical results of lid-driven cavity flow with the Reynolds number Re = 5,000, 7,500 and 10,000. From these numerical results, we can see that the schemes can result in good accuracy, which shows that this method is highly efficient.     

An averaging scheme for macroscopic numerical simulation of nonconvex minimization problems

Averaging or gradient recovery techniques, which are a popular tool for improved convergence or superconvergence of finite element methods in elliptic partial differential equations, have not been recommended for nonconvex minimization problems as the energy minimization process enforces finer and finer oscillations and hence at the first glance, a smoothing step appears even counterproductive. For macroscopic quantities such as the stress field, however, this counterargument is no longer true. In fact, this paper advertises an averaging technique for a surprisingly improved convergence behavior for nonconvex minimization problems. Similar to a finite volume scheme, numerical experiments on a double-well benchmark example provide empirical evidence of superconvergence phenomena in macroscopic numerical simulations of oscillating microstructures. AMS subject classification (2000)  65K10,65N30     

Superlinearly convergent CG methods via equivalent preconditioning for nonsymmetric elliptic operators

Summary. The convergence of the conjugate gradient method is studied for preconditioned linear operator equations with nonsymmetric normal operators, with focus on elliptic convection-diffusion operators in Sobolev space. Superlinear convergence is proved first for equations whose preconditioned form is a compact perturbation of the identity in a Hilbert space. Then the same convergence result is verified for elliptic convection-diffusion equations using different preconditioning operators. The convergence factor involves the eigenvalues of the corresponding operators, for which an estimate is also given. The above results enable us to verify the mesh independence of the superlinear convergence estimates for suitable finite element discretizations of the convection-diffusion problems.Mathematics Subject Classification (2000): 65J10, 65F10, 65N15The second author was supported by the Hungarian Research Grant OTKA No. T. 043765.Dedicated to David M. Young on the occasion of his 80th birthday.     

Sufficient conditions for uniform convergence on layer-adapted meshes for one-dimensional reaction–diffusion problems

We study convergence properties of a finite element method with lumping for the solution of linear one-dimensional reaction–diffusion problems on arbitrary meshes. We derive conditions that are sufficient for convergence in the L_∞ norm, uniformly in the diffusion parameter, of the method. These conditions are easy to check and enable one to immediately deduce the rate of convergence. The key ingredients of our analysis are sharp estimates for the discrete Green function associated with the discretization. AMS subject classification 65L10, 65L12, 65L15     

A finite element,filtered eddy-viscosity method for the Navier–Stokes equations with large Reynolds number

The direct numerical simulation of the Navier–Stokes system in turbulent regimes is a formidable task due to the disparate scales that have to be resolved. Turbulence modeling attempts to mitigate this situation by somehow accounting for the effects of small-scale behavior on that at large-scales, without explicitly resolving the small scales. One such approach is to add viscosity to the problem; the Smagorinsky and Ladyzhenskaya models and other eddy-viscosity models are examples of this approach. Unfortunately, this approach usually results in over-dampening at the large scales, i.e., large-scale structures are unphysically smeared out. To overcome this fault of simple eddy-viscosity modeling, filtered eddy-viscosity methods that add artificial viscosity only to the high-frequency modes were developed in the context of spectral methods. We apply the filtered eddy-viscosity idea to finite element methods based on hierarchical basis functions. We prove the existence and uniqueness of the finite element approximation and its convergence to solutions of the Navier–Stokes system; we also derive error estimates for finite element approximations.     

Hamilton-Jacobi方程的单调有限元格式

A monotone finite element scheme is obtained by applying the finite element method to the viscosity equation of the Hamilton-Jacobi equation on unstructured meshes. Under some constraints, we show that this scheme is monotone and its numerical solution converges to the viscosity solution of the Hamilton-Jacobi equa-tion. Numerical examples test the stability and the convergence of this scheme.     

On the convergence of basic iterative methods for convection–diffusion equations

In this paper we analyze convergence of basic iterative Jacobi and Gauss–Seidel type methods for solving linear systems which result from finite element or finite volume discretization of convection–diffusion equations on unstructured meshes. In general the resulting stiffness matrices are neither M‐matrices nor satisfy a diagonal dominance criterion. We introduce two newmatrix classes and analyse the convergence of the Jacobi and Gauss–Seidel methods for matrices from these classes. A new convergence result for the Jacobi method is proved and negative results for the Gauss–Seidel method are obtained. For a few well‐known discretization methods it is shown that the resulting stiffness matrices fall into the new matrix classes. Copyright © 1999 John Wiley & Sons, Ltd.     

求解椭圆边值问题惩罚形式的间断有限元方法

本文提出了一个新的求解二阶椭圆边值问题的惩罚形式间断有限元方法并给出了稳定性和收敛性分析. 特别地,本文建立了间断有限元解的基于余量的后验误差估计,给出了求解间断有限元方程的自适应算法.       

A primal mixed formulation for exterior transmission problems in ${\bf R}^2$

Summary.   We combine a primal mixed finite element approach with a Dirichlet-to-Neumann mapping (arising from the boundary integral equation method) to study the weak solvability and Galerkin approximations of a class of linear exterior transmission problems in potential theory. Our results are mainly based on the Babuska-Brezzi theory for variational problems with constraints. We establish the uniqueness of solution for the continuous and discrete formulations, and show that finite element subspac es of Lagrange type satisfy the discrete compatibility conditions. In addition, we provide the error analysis, including polygonal approximations of the domain, and prove strong convergence of the Galerkin solutions. Moreover, under additional regularity assumptions on the solution of the continuous formulation, we obtain the asymptotic rate of convergence O(h). Received August 25, 1998 / Revised version received March 8, 2000 / Published online October 16, 2000     

A posteriori error estimates by recovered gradients in parabolic finite element equations

This paper considers a posteriori error estimates by averaged gradients in second order parabolic problems. Fully discrete schemes are treated. The theory from the elliptic case as to when such estimates are asymptotically exact, on an element, is carried over to the error on an element at a given time. The basic principle is that the elliptic theory can be extended to the parabolic problems provided the time-step error is smaller than the space-discretization error. Numerical illustrations confirming the theoretical results are given. Our results are not practical in the sense that various constants can not be estimated realistically. They are conceptual in nature. AMS subject classification (2000)  65M60, 65M20, 65M15     

非定常线性化Navier-Stokes方程的子格粘性非协调有限元方法

本文结合子格粘性法的思想,空间采用非协调Crouzeix-Raviart元逼近,时间采用Crank-Nicolson差分离散,对非定常线性化Navier-Stokes方程建立了全离散的子格粘性非协调有限元格式.对稳定性和误差估计作出了详细的分析, 得出了最优的误差估计.最后, 通过数值算例进一步验证了该方法的稳定性和收敛性.     

Superconvergence of finite element discretization of time relaxation models of advection

The nodal accuracy of finite element discretizations of advection equations including a time relaxation term is studied. Worst case error estimates have been proven for this combination (for the Navier–Stokes equations) by energy methods. By considering the Cauchy problem with uniform meshes, precise Fourier analysis of the error is possible. This analysis shows (1) the worst case upper bounds are sharp in the meshwidth h, (2) time relaxation stabilization does not degrade superconvergence of the usual FEM, (3) time relaxation itself is possibly superconvergent and (4) higher order time relaxation is preferable to maintain small numerical errors. AMS subject classification (2000)  primary 65M60, secondary 65M15