Monotone iterative methods for finite difference system of reaction-diffusion equations

Summary This paper presents an existence-comparison theorem and an iterative method for a nonlinear finite difference system which corresponds to a class of semilinear parabolic and elliptic boundary-value problems. The basic idea of the iterative method for the computation of numerical solutions is the monotone approach which involves the notion of upper and lower solutions and the construction of monotone sequences from a suitable linear discrete system. Using upper and lower solutions as two distinct initial iterations, two monotone sequences from a suitable linear system are constructed. It is shown that these two sequences converge monotonically from above and below, respectively, to a unique solution of the nonlinear discrete equations. This formulation leads to a well-posed problem for the nonlinear discrete system. Applications are given to several models arising from physical, chemical and biological systems. Numerical results are given to some of these models including a discussion on the rate of convergence of the monotone sequences.     

A discrete solenoidal finite difference scheme for the numerical approximation of incompressible flows

Summary For a well known class of finite difference schemes for approximating incompressible flows it is shown that the condition of discrete incompressibility can be incorporated into the discrete space. This simplifies the structure of the linear or nonlinear discrete systems and reduces the number of unknowns.     

On finite element methods for the Neumann problem

Summary The Neumann problem for a second order elliptic equation with self-adjoint operator is considered, the unique solution of which is determined from projection onto unity. Two variational formulations of this problem are studied, which have a unique solution in the whole space. Discretization is done via the finite element method based on the Ritz process, and it is proved that the discrete solutions converge to one of the solutions of the continuous problem. Comparison of the two methods is done.     

Estimation of convergence orders in repeated richardson extrapolation

LetA(h) be an approximation depending on a single parameterh to a fixed quantityA, and assume thatA–A(h)=c ₁ h k ₁ +c ₂ h ^k ₂ +.... Given a sequence ofh-valuesh ₁>h ₂>...>h _n and corresponding computed approximationsA(h _i ), the orders for repeated Richardson extrapolation are estimated, and the repeated extrapolation is performed. Hence in this case the algorithm described here can do the same work as Brezinski'sE-algorithm and at the same time provide a check whether repeated extrapolation is justified.     

Defect correction for nonlinear elliptic difference equations

Summary The present paper is concerned with the study of a high-order defect correction technique for discretizations of nonlinear elliptic boundary value problems. The convergence of the method is analyzed in general and, in more detail, for a specific example. The algorithmic combination of defect correction and multigrid techniques is also discussed.This work was supported by Österreichischer Fonds zur Förderung der wissenschaftlichen Forschung     

On the construction of optimal mixed finite element methods for the linear elasticity problem

Summary The mixed finite element method for the linear elasticity problem is considered. We propose a systematic way of designing methods with optimal convergence rates for both the stress tensor and the displacement. The ideas are applied in some examples.     

A relaxation procedure for domain decomposition methods using finite elements

Summary We present the convergence analysis of a new domain decomposition technique for finite element approximations. This technique was introduced in [11] and is based on an iterative procedure among subdomains in which transmission conditions at interfaces are taken into account partly in one subdomain and partly in its adjacent. No global preconditioner is needed in the practice, but simply single-domain finite element solvers are required. An optimal strategy for an automatic selection of a relaxation parameter to be used at interface subdomains is indicated. Applications are given to both elliptic equations and incompressible Stokes equations.     

A family of higher order mixed finite element methods for plane elasticity

Summary The Dirichler problem for the equations of plane elasticity is approximated by a mixed finite element method using a new family of composite finite elements having properties analogous to those possessed by the Raviart-Thomas mixed finite elements for a scalar, second-order elliptic equation. Estimates of optimal order and minimal regularity are derived for the errors in the displacement vector and the stress tensor inL ²(), and optimal order negative norm estimates are obtained inH ^s () for a range ofs depending on the index of the finite element space. An optimal order estimate inL () for the displacement error is given. Also, a quasioptimal estimate is derived in an appropriate space. All estimates are valid uniformly with respect to the compressibility and apply in the incompressible case. The formulation of the elements is presented in detail.This work was performed while Professor Arnold was a NATO Postdoctoral Fellow     

Approximation by finite differences of the propagation of acoustic waves in stratified media

Summary In this paper, we analyze the approximation of acoustic waves in a two layered media by a finite diffrences variational scheme. We examine in particular the approximation of the guided waves. We point out the existence of purely numerical parasitic phenomena and quantify the numerical dispersion relative to guided waves.     

On some boundary element methods for the heat equation

Summary The object of this paper is to study some boundary element methods for the heat equation. Two approaches are considered. The first, based on the heat potential, has been studied numerically by previous authors. Here the convergence analysis in one space dimension is presented. In the second approach, the heat equation is first descretized in time and the resulting elliptic problem is put in the boundary formulation. A straight forward implicit method and Crank-Nicolson's method are thus studied. Again convergence in one space dimension is proved.     

A contour integral representation of linear extrapolation methods

Summary Extrapolation methods are well-known to be very efficient tools for the acceleration of the convergence of certain sequences of numbers or functions, cf. [2, 3, 4, 8, 9]. In this note we present a representation of linear extrapolation procedures in terms of complex contour integrals. For the proof we make use of a complete characterization of these procedures as linear functionals, which is itself of some interest and can be found, for example, in [2] or [8].     

Linearly implicit extrapolation methods for differential-algebraic systems

Summary We consider the numerical solution of implicit differential equations in which the solution derivative appears multiplied by a solution-dependent singular matrix. We study extrapolation methods based on two linearly implicit Euler discretizations. Their error behaviour is explained by perturbed asymptotic expansions.     

On the numerical approximation of finite speed diffusion problems

Summary Lagrangian formulations for the Cauchy problems for the generalized-heat and porous-media equations are introduced and equivalence and existence results discussed. Efficient interface tracking finite difference and finite element discretizations of the Lagrangian formulation are discussed. Mixed Euler-Lagrange formulations for mixed problems and the one phase Stefan problem are presented. Numerical experiments are discussed.Dedicated on the occasion of Prof. Ivo Babuka's 60th birthday     

One-step and extrapolation methods for differential-algebraic systems

Summary The paper analyzes one-step methods for differential-algebraic equations (DAE) in terms of convergence order. In view of extrapolation methods, certain perturbed asymptotic expansions are shown to hold. For the special DAE extrapolation solver based on the semi-implicit Euler discretization, the perturbed order pattern of the extrapolation tableau is derived in detail. The theoretical results lead to modifications of the known code. The efficiency of the modifications is illustrated by numerical comparisons over critical examples mainly from chemical combustion.     

On finite element subspaces over quadrilateral and hexahedral meshes for incompressible viscous flow problems

Summary Optimal rates of convergence for the approximate solution of the stationary Stokes equations are obtained for finite element schemes which use piecewise constants to approximate the pressure and piecewise linear or piecewise bilinear or trilinear polynomials to approximate the velocity over fairly general quadrilateral or hexahedral meshes.This research was supported by NSF Grant MC80-16532     

Superconvergence for rectangular mixed finite elements

Summary In this paper we prove superconvergence error estimates for the vector variable for mixed finite element approximations of second order elliptic problems. For the rectangular finite elements of Raviart and Thomas [19] and for those of Brezzi et al. [4] we prove that the distance inL ² between the approximate solution and a projection of the exact one is of higher order than the error itself.This result is exploited to obtain superconvergence at Gaussian points and to construct higher order approximations by a local postprocessing.     

On the finite volume element method

Summary The finite volume element method (FVE) is a discretization technique for partial differential equations. It uses a volume integral formulation of the problem with a finite partitioning set of volumes to discretize the equations, then restricts the admissible functions to a finite element space to discretize the solution. this paper develops discretization error estimates for general selfadjoint elliptic boundary value problems with FVE based on triangulations with linear finite element spaces and a general type of control volume. We establishO(h) estimates of the error in a discreteH ¹ semi-norm. Under an additional assumption of local uniformity of the triangulation the estimate is improved toO(h ²). Results on the effects of numerical integration are also included.This research was sponsored in part by the Air Force Office of Scientific Research under grant number AFOSR-86-0126 and the National Science Foundation under grant number DMS-8704169. This work was performed while the author was at the University of Colorado at Denver     

On a generalization of the Richardson extrapolation process

Summary A convergence result for a generalized Richardson extrapolation process is improved upon considerably and additional results of interest are proved. An application of practical importance is also given. Finally, some known results concerning the convergence of Levin's T-transformation are reconsidered in light of the results of the present work.