Error analysis of fully discrete velocity-correction methods for incompressible flows

A fully discrete version of the velocity-correction method, proposed by Guermond and Shen (2003) for the time-dependent Navier-Stokes equations, is introduced and analyzed. It is shown that, when accounting for space discretization, additional consistency terms, which vanish when space is not discretized, have to be added to establish stability and optimal convergence. Error estimates are derived for both the standard version and the rotational version of the method. These error estimates are consistent with those by Guermond and Shen (2003) as far as time discretiztion is concerned and are optimal in space for finite elements satisfying the inf-sup condition.

     

A comparative study of null‐space factorizations for sparse symmetric saddle point systems

Null‐space methods for solving saddle point systems of equations have long been used to transform an indefinite system into a symmetric positive definite one of smaller dimension. A number of independent works in the literature have identified that we can interpret a null‐space method as a matrix factorization. We review these findings, highlight links between them, and bring them into a unified framework. We also investigate the suitability of using null‐space factorizations to derive sparse direct methods and present numerical results for both practical and academic problems.     

New splitting methods for two-dimensional evolutionary equations

New second- and third-order splitting methods are proposed for evolutionary-type partial differential equations in a two-dimensional space. These methods are derived on the basis of diagonally implicit methods applied to the numerical analysis of stiff ordinary differential equations. The splitting methods are found to be absolutely unconditionally stable. Test calculations are presented.     

耦合有限元空间上Poisson方程的广义差分法

本文讨论了耦合有限元空间上Poisson方程的一种广义差分法 ,试探函数空间为耦合的有限元空间 ,检验函数空间为与对偶单元相对应的分片常数空间 ,并给出了误差估计 .     

A family of variable metric methods in function space,without exact line searches

The purpose of this paper is to extend a family of variable metric methods, of which the BFGS algorithm (Ref. 1) is a member, into function space, in particular, for the solution of unconstrained optimal control problems. An inexact one-dimensional minimization as suggested by Fletcher (ref. 2) is used. It is shown that, with this stepsize rule and under some mild assumptions, the sequence constructed by this family of methods converges superlinearly for a strictly convex functional defined on a suitable Banach space. This result is shown to remain valid on a Hilbert space and on a Euclidean space under more relaxed assumptions. The BFGS method without line searches is used to solve several standard numerical examples, and excellent performance is observed.This work was supported by the Consejo Nacional de Ciencia y Tecnologia de Mexico, and by the National Research Council of Canada, Grant No. A-8835. The authors are indebted to Dr. C. Charalambous for suggesting the topic and stimulating discussions.     

IMPLICIT-EXPLICIT MULTISTEP FINITE ELEMENT-MIXED FINITE ELEMENT METHODS FOR THE TRANSIENT BEHAVIOR OF A SEMICONDUCTOR DEVICE

The transient behavior of a semiconductor device consists of a Poisson equa-tion for the electric potential and of two nonlinear parabolic equations for the electrondensity and hole density.The electric potential equation is discretized by a mixed finiteelement method. The electron and hole density equations are treated by implicit-explicitmultistep finite element methods. The schemes are very efficient. The optimal order errorestimates both in time and space are derived.     

New fourth-order splitting methods for two-dimensional evolution equations

New second- and third-order splitting methods are proposed for partial differential equations of the evolution type in a two-dimensional space. The methods are derived as based on diagonal implicit techniques used in the numerical solution to stiff ordinary differential equations. The methods are absolutely and unconditionally stable. Test computations are presented.     

Arbitrary-order functionally fitted energy-diminishing methods for gradient systems

It is well known that for gradient systems in Euclidean space or on a Riemannian manifold, the energy decreases monotonically along solutions. In this letter we derive and analyse functionally fitted energy-diminishing methods to preserve this key property of gradient systems. It is proved that the novel methods are energy-diminishing and can achieve damping for very stiff gradient systems. We also show that the methods can be of arbitrarily high order and discuss their implementations. A numerical test is reported to illustrate the efficiency of the new methods in comparison with three existing numerical methods in the literature.     

Numerical Verification Methods for Solutions of the Free Boundary Problem

We propose two methods to enclose the solution of an ordinary free boundary problem. The problem is reformulated as a nonlinear boundary value problem on a fixed interval including an unknown parameter. By appropriately setting a functional space that depends on the finite element approximation, the solution is represented as a fixed point of a compact map. Then, by using the finite element projection with constructive error estimates, a Newton-type verification procedure is derived. In addition, numerical examples confirming the effectiveness of current methods are given.     

Robust Schur complement method for strongly anisotropic elliptic equations

We present robust and asymptotically optimal iterative methods for solving 2D anisotropic elliptic equations with strongly jumping coefficients, where the direction of anisotropy may change sharply between adjacent subdomains. The idea of a stable preconditioning for the Schur complement matrix is based on the use of an exotic non‐conformal coarse mesh space and on a special clustering of the edge space components according to the anisotropy behavior. Our method extends the well known BPS interface preconditioner [2] to the case of anisotropic equations. The technique proposed also provides robust solvers for isotropic equations in the presence of degenerate geometries, in particular, in domains composed of thin substructures. Numerical experiments confirm efficiency and robustness of the algorithms for the complicated problems with strongly varying diffusion and anisotropy coefficients as well as for the isotropic diffusion equations in the ‘brick and mortar’ structures involving subdomains with high aspect ratios. Copyright © 1999 John Wiley & Sons, Ltd.     

Neumann Control of Unstable Parabolic Systems: Numerical Approach

The present article is concerned with the Neumann control of systems modeled by scalar or vector parabolic equations of reaction-advection-diffusion type with a particular emphasis on systems which are unstable if uncontrolled. To solve these problems, we use a combination of finite-difference methods for the time discretization, finite-element methods for the space discretization, and conjugate gradient algorithms for the iterative solution of the discrete control problems. We apply then the above methodology to the solution of test problems in two dimensions, including problems related to nonlinear models.     

Branch-and-Bound Variant of an Outcome-Based Algorithm for Optimizing over the Efficient Set of a Bicriteria Linear Programming Problem

The paper presents a finite branch-and-bound variant of an outcome-based algorithm proposed by Benson and Lee for minimizing a lower-semicontinuous function over the efficient set of a bicriteria linear programming problem. Similarly to the Benson-Lee algorithm, we work primarily in the outcome space. Dissimilarly, instead of constructing a sequence of consecutive efficient edges in the outcome space, we use the idea of generating a refining sequence of partitions covering the at most two-dimensional efficient set in the outcome space. Computational experience is also presented.     

Fixed-bed drying simulation of agricultural products using a new backward finite difference scheme

This work is concerned with the numerical simulation of fixed-bed corn drying using MSU (Michigan State University) drying model. The classical numerical procedure for MSU model relies on an explicit method of finite differences which requires certain stability conditions between the step sizes of the time and space variables. The objective of the present paper is to establish a stable implicit method based on backward finite differences, in both time and space variables, which takes into account some specific empirical aspects of the problem. Computational results illustrate the efficiency and the flexibility of method.     

An Eulerian‐Lagrangian control volume scheme for two‐dimensional unsteady advection‐diffusion problems

We develop a mass conservative Eulerian‐Lagrangian control volume scheme (ELCVS) for the solution of the transient advection‐diffusion equations in two space dimensions. This method uses finite volume test functions over the space‐time domain defined by the characteristics within the framework of the class of Eulerian‐Lagrangian localized adjoint characteristic methods (ELLAM). It, therefore, maintains the advantages of characteristic methods in general, and of this class in particular, which include global mass conservation as well as a natural treatment of all types of boundary conditions. However, it differs from other methods in that class in the treatment of the mass storage integrals at the previous time step defined on deformed Lagrangian regions. This treatment is especially attractive for orthogonal rectangular Eulerian grids composed of block elements. In the algorithm, each deformed region is approximated by an eight‐node region with sides drawn parallel to the Eulerian grid, which significantly simplifies the integration compared with the approach used in finite volume ELLAM methods, based on backward tracking, while retaining local mass conservation at no additional expenses in terms of accuracy or CPU consumption. This is verified by numerical tests which show that ELCVS performs as well as standard finite volume ELLAM methods, which have previously been shown to outperform many other well‐received classes of numerical methods for the equations considered. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2012     

New least-square algorithms

New algorithms are presented for approximating the minimum of the sum of squares ofM real and differentiable functions over anN-dimensional space. These algorithms update estimates for the location of a minimum after each one of the functions and its first derivatives are evaluated, in contrast with other least-square algorithms which evaluate allM functions and their derivatives at one point before using any of this information to make an update. These new algorithms give estimates which fluctuate about a minimum rather than converging to it. For many least-square problems, they give an adequate approximation for the solution more quickly than do other algorithms.It is a pleasure to thank J. Chesick of Haverford College for suggesting and implementing the application of this algorithm to x-ray crystallography.     

A steepest-descent method for optimization of mechanical systems

Steepest-descent optimal control techniques have been used extensively for dynamic systems in one independent variable and with a full set of initial conditions. This paper presents an extension of the steepest-descent technique to mechanical design problems that are described by boundary-value problems with one or more independent variables. The method is illustrated by solving finite-dimensional problems, problems with distribution of design over one space dimension, and problems with distribution of design over two space dimensions.     

Parabolic target space and primal-dual interior-point methods

In this paper we develop new primal-dual interior-point methods for linear programming problems, which are based on the concept of parabolic target space. We show that such schemes work in the infinity-neighborhood of the primal-dual central path. Nevertheless, these methods possess the best known complexity estimate. We demonstrate that the adaptive-step path-following strategies can be naturally incorporated in such schemes.     

Reference point approximation method for the solution of bicriterial nonlinear optimization problems

This paper presents a reference point approximation algorithm which can be used for the interactive solution of bicriterial nonlinear optimization problems with inequality and equality constraints. The advantage of this method is that the decision maker may choose arbitrary reference points in the criteria space. Moreover, a special tunneling technique is given for the computation of global solutions of certain subproblems. Finally, the proposed method is applied to a mathematical example and a problem in mechanical engineering.     

Hybrid pseudospectral–finite difference method for solving a 3D heat conduction equation in a submicroscale thin film

This research aims to develop a time‐dependent pseudospectral‐finite difference scheme for solving a 3D dual‐phase‐lagging heat transport equation in a submicroscale thin film. The scheme uses periodic pseudospectral discretization in space and a fully second‐order finite difference discretization in time. The three consecutive time steps model is then solved explicitly, by using a preconditioned conjugate gradient method. The scheme is illustrated by an example which is used to investigate the heat transfer in a gold submicroscale thin film. Comparisons are made with available literature. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007