求解一类隐式互补问题的加速模系矩阵分裂迭代方法

本文提出求解一类隐式互补问题的加速模系矩阵分裂迭代法.通过将隐式互补问题重新表述为一个等价的不动点方程,建立一类新的基于模系的两步矩阵分裂方法,并在一定条件下证明了方法的收敛性.数值实验表明,该方法在迭代步数上优于传统的模系矩阵分裂迭代方法.     

求解一类隐线性互补问题的投影MAOR算法

用MAOR迭代算法求解一类L-矩阵的隐线性互补问题.证明了由此算法产生的迭代序列的聚点是隐线性互补问题的解.并且当问题中的矩阵是M-矩阵时,算法产生的迭代序列单调收敛于隐互补问题的解.     

求解障碍问题自由边界的一种非精确光滑方法

利用差分原理将一类数学物理障碍问题转化为线性互补问题.给出了求解大规模线性互补问题的一种非精确光滑算法,证明了该算法的适定性和全局收敛性.数值试验表明该方法能很好地求解此类障碍问题.     

不可压Navier-Stokes方程的隐式投影法

起的解的小尺度以外,还有解的约束条件,即(1.1)。为了分辨小尺度,需用足够小的网格;而为了保证计算效率,时间步△_t需适当地大,从而必须用隐式格式。但是,由于解的约束条件,隐式格式的实现有困难,为此可用所谓的投影法,如[1]-[3]。前两者基于全隐式或Crank-Nicholson(CN)隐式和所谓的全投影;后者基于CN和压力修正投影。当然可以同时迭代u,v,p而得出每时间层的解,如[4]中的Peyret以及[5]中的Spolding-     

一类求解单调变分不等式的隐式方法

1．引言变分不等式是一个非常有趣。非常困难的数学问题［"］．它具有广泛的应用（例如，数学规划中的许多基本问题都可以归结为一个变分不等式问题），因而得到深入的研究并有了不少算法［1，2，5－8，17－21］．对线性单调变分不等式，我们最近提出了一系列投影收缩算法Ig－13］．本文考虑求解单调变分不等式其中0CW是一闭凸集，F是从正p到自身的一个单调算子，一即有我们用比（·）表示到0上的投影．求解单调变分不等式的一个简单方法是基本投影法［1，6］，它的迭代式为然而，如果F不是仿射函数，只有当F一致强单调且LIPSChitZ连续…     

求解非线性不适定问题的隐式迭代法

将处理线性不适定算子方程的隐式迭代法推广到非线性不适定问题,证明了迭代解误差序列的单调性,并进一步利用迭代误差的单调性得出求解非线性不适定问题隐式迭代法对精确方程和扰动方程的收敛性.     

延迟微分方程隐式Euler方法的收敛性

本文讨论了用隐式Euler方法求解一类延迟量满足Lipschitz条件且Lipschitz常数小于1的非线性变延迟微分方程初值问题的收敛性.获得了带线性插值的隐式Euler方法的收敛性结果.     

求解隐式差分方程的加速迭代并行算法

1、引言 近年来,求解抛物型方程的有限差分并行迭代算法有了较大发展．针对稳定性好且难于并行化的隐式差分方程,文第一次提出了构造分段隐式的思想,建立了分段显-隐式（ASE-Ⅰ）方法和交替分段Crank-Nicolson（ASC-N）方法,实现了分而治之原则,     

一种半隐式有限体积—有限元方法的收敛性

本文研究非线性对流扩散问题的一种半隐式有限体积和有限元方法相结合的数值方法,给出数值解的收敛性及其证明。     

Numerical Approximation of a Unilateral Obstacle Problem

We consider a reformulation of the unilateral obstacle problem presented by the authors (Addou and Mermri in Math-Rech. Appl. 2:59–69, 2000). This reformulation introduces a continuous function, whose subdifferential characterizes the noncontact domain. Our goal in this paper is to give a numerical approximation of the solution of the reformulated problem. We consider discretization of the problem based on finite element method. Then we prove the convergence of the approximate solution to the exact one. Some numerical tests on one-dimensional obstacle problem are provided.     

Implicit Interval Methods for Solving the Initial Value Problem

Implicit interval methods of Runge–Kutta and Adams–Moulton type for solving the initial value problem are proposed. It can be proved that the exact solution of the problem belongs to interval-solutions obtained by the considered methods. Furthermore, it is possible to estimate the widths of interval-solutions.     

Linearly Implicit Finite Element Methods for the Time-Dependent Joule Heating Problem

Completely discrete numerical methods for a nonlinear elliptic-parabolic system, the time-dependent Joule heating problem, are introduced and analyzed. The equations are discretized in space by a standard finite element method, and in time by combinations of rational implicit and explicit multistep schemes. The schemes are linearly implicit in the sense that they require, at each time level, the solution of linear systems of equations. Optimal order error estimates are proved under the assumption of sufficiently regular solutions. AMS subject classification (2000) 65M30, 65M15, 35K60     

An Iteration Scheme for Implicit Runge--Kutta Methods

In the implementation of implicit Runge-Kutta methods the computationaleffort is typically dominated by the cost of solving large setsof non-linear equations. As an alternative to the usual Newtonmethod, an iteration scheme is considered that sacrifices superlinearconvergence for reduced linear algebra costs. By modifying thisnew scheme, using the successive over-relaxation technique,an improvement is achieved in the rate of convergence.     

An Obstacle Problem for Nonlocal Equations in Perforated Domains

In this paper we analyze the behavior of solutions to a nonlocal equation of the form J ? u (x) ? u (x) = f (x) in a perforated domain Ω ? A ^?? with u = 0 in \(A^{\epsilon } \cup {\Omega }^{c}\) and an obstacle constraint, u ≥ ψ in Ω ? A ^?? . We show that, assuming that the characteristic function of the domain Ω ? A ^?? verifies \(\chi _{\epsilon } \rightharpoonup \mathcal {X}\) weakly ^? in \(L^{\infty }({\Omega })\), there exists a weak limit of the solutions u ^?? and we find the limit problem that is satisfied in the limit. When \(\mathcal {X} \not \equiv 1\) in this limit problem an extra term appears in the equation as well as a modification of the obstacle constraint inside the domain.     

An Obstacle Control Problem with a Source Term

   Abstract. An optimal control problem for an elliptic variational inequality with a source term is considered. The obstacle is the control, and the goal is to keep the solution of the variational inequality close to the desired profile while the H ¹ norm of the obstacle is not too large. The addition of the source term strongly affects the needed compactness result for the existence of a minimizer.     

An Obstacle Control Problem with a Source Term

Abstract. An optimal control problem for an elliptic variational inequality with a source term is considered. The obstacle is the control, and the goal is to keep the solution of the variational inequality close to the desired profile while the H ¹ norm of the obstacle is not too large. The addition of the source term strongly affects the needed compactness result for the existence of a minimizer.     

Two-Phase Obstacle Problem

On the basis of the monotonicity formula due to Alt, Caffarelli, and Friedman, the boundedness of the second-order derivatives D ² u of solutions to the equation

A Projection Preconditioner for Solving the Implicit Immersed Boundary Equations

This paper presents a method for solving the linear semi-implicit immersed boundary equations which avoids the severe time step restriction presented by explicit-time methods. The Lagrangian variables are eliminated via a Schur complement to form a purely Eulerian saddle point system, which is preconditioned by a projection operator and then solved by a Krylov subspace method. From the viewpoint of projection methods, we derive an ideal preconditioner for the saddle point problem and compare the efficiency of a number of simpler preconditioners that approximate this perfect one. For low Reynolds number and high stiffness, one particular projection preconditioner yields an efficiency improvement of the explicit IB method by a factor around thirty. Substantial speed-ups over explicit-time method are achieved for Reynolds number below 100. This speedup increases as the Eulerian grid size and/or the Reynolds number are further reduced.