On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs

Summary The Schwarz Alternating Method can be used to solve elliptic boundary value problems on domains which consist of two or more overlapping subdomains. The solution is approximated by an infinite sequence of functions which results from solving a sequence of elliptic boundary value problems in each subdomain. In this paper, proofs of convergence of some Schwarz Alternating Methods for nonlinear elliptic problems which are known to have solutions by the monotone method (also known as the method of subsolutions and supersolutions) are given. In particular, an additive Schwarz method for scalar as well some coupled nonlinear PDEs are shown to converge to some solution on finitely many subdomains, even when multiple solutions are possible. In the coupled system case, each subdomain PDE is linear, decoupled and can be solved concurrently with other subdomain PDEs. These results are applicable to several models in population biology. This work was in part supported by a grant from the RGC of HKSAR, China (HKUST6171/99P)     

Composite Laguerre-Legendre spectral method for exterior problems

In this paper, we propose a composite Laguerre-Legendre spectral method for two-dimensional exterior problems. Results on the composite Laguerre-Legendre approximation, which is a set of piecewise mixed approximations coupled with domain decomposition, are established. These results play important roles in the related spectral methods for exterior problems. As examples of applications, the composite spectral schemes are provided for two model problems, with the convergence analysis. An efficient implementation is described. Numerical results demonstrate the spectral accuracy in space of this new approach, and confirm the analysis. The approximation results and techniques developed in this paper are also applicable to other problems defined on unbounded domains.     

Numerical analysis of the coupling of free fluid with a poroelastic material

This paper presents a numerical solution of the coupled system of the time-dependent Stokes and fully dynamic Biot equations. The numerical scheme is based on standard inf-sup stable finite elements in space and the Backward Euler scheme in time. We establish stability of the scheme and derive error estimates for the fully discrete coupled scheme. To handle realistic parameters which may cause nonphysical oscillations in the pore fluid pressure, a heuristic stabilization technique is considered. Numerical errors and convergence rates for smooth problems as well as tests on realistic material parameters are presented.     

Existence of a solution for two phase flow in porous media: The case that the porosity depends on the pressure

In this paper we prove the existence of a solution of a coupled system involving a two phase incompressible flow in the ground and the mechanical deformation of the porous medium where the porosity is a function of the global pressure. The model is strongly coupled and involves a nonlinear degenerate parabolic equation. In order to show the existence of a weak solution, we consider a sequence of related uniformly parabolic problems and apply the Schauder fixed point theorem to show that they possess a classical solution. We then prove the relative compactness of sequences of solutions by means of the Fréchet-Kolmogorov theorem; this yields the convergence of a subsequence to a weak solution of the parabolic system.     

Iterative Methods for a Linearly Perturbed Algebraic Matrix Riccati Equation Arising in Stochastic Control

We start with a discussion of coupled algebraic Riccati equations arising in the study of linear-quadratic optimal control problems for Markov jump linear systems. Under suitable assumptions, this system of equations has a unique positive semidefinite solution, which is the solution of practical interest. The coupled equations can be rewritten as a single linearly perturbed matrix Riccati equation with special structures. We study the linearly perturbed Riccati equation in a more general setting and obtain a class of iterative methods from different splittings of a positive operator involved in the Riccati equation. We prove some special properties of the sequences generated by these methods and determine and compare the convergence rates of these methods. Our results are then applied to the coupled Riccati equations of jump linear systems. We obtain linear convergence of the Lyapunov iteration and the modified Lyapunov iteration, and confirm that the modified Lyapunov iteration indeed has faster convergence than the original Lyapunov iteration.     

Short shots and industrial case studies: Understanding fluid flow and solidification in high pressure die casting

The geometric complexity and high fluid speeds involved in high pressure die casting (HPDC) combine to give strongly three dimensional fluid flow with significant free surface fragmentation and splashing. A simulation method that has proved particularly suited to modelling HPDC is Smoothed Particle Hydrodynamics (SPH). Materials are approximated by particles that are free to move around rather than by fixed grids, enabling more accurate prediction of fluid flows involving complex free surface motion. Three practical industrial case studies of SPH simulated HPDC flows are presented; aluminium casting of a differential cover (automotive), an electronic housing and zinc casting of a door lock plate. These show significant detail in the fragmented fluid free surfaces and allow us to understand the predisposition to create defects such as porosity in the castings. The validation of flow predictions coupled with heat transfer and solidification is an important area for such modelling. One powerful approach is to use short shots, where insufficient metal is used in the casting or the casting shot is halted part way through, to leave the die cavity only partially filled. The frozen partial castings capture significant detail about the order of fill and the flow structures occurring during different stages of filling. Validation can occur by matching experimental and simulated short shots. Here we explore the effect of die temperature, metal super-heat and volume fill on the short shots for the casting of a simple coaster. The bulk features of the final solid castings are found to be in good agreement with the predictions, but the fine details appear to depend on surface behaviour of the solidifying metals. This potentially has significant implications for modelling HPDC.     

Solutions for the Linear-Quadratic Control Problem of Markov Jump Linear Systems

The paper is concerned with recursive methods for obtaining the stabilizing solution of coupled algebraic Riccati equations arising in the linear-quadratic control of Markovian jump linear systems by solving at each iteration uncoupled algebraic Riccati equations. It is shown that the new updates carried out at each iteration represent approximations of the original control problem by control problems with receding horizon, for which some sequences of stopping times define the terminal time. Under this approach, unlike previous results, no initialization conditions are required to guarantee the convergence of the algorithms. The methods can be ordered in terms of number of iterations to reach convergence, and comparisons with existing methods in the current literature are also presented. Also, we extend and generalize current results in the literature for the existence of the mean-square stabilizing solution of coupled algebraic Riccati equations.     

A method for solving an exterior three-dimensional boundary value problem for the Laplace equation

We develop and experimentally study the algorithms for solving three-dimensionalmixed boundary value problems for the Laplace equation in unbounded domains. These algorithms are based on the combined use of the finite elementmethod and an integral representation of the solution in a homogeneous space. The proposed approach consists in the use of the Schwarz alternating method with consecutive solution of the interior and exterior boundary value problems in the intersecting subdomains on whose adjoining boundaries the iterated interface conditions are imposed. The convergence of the iterative method is proved. The convergence rate of the iterative process is studied analytically in the case when the subdomains are spherical layers with the known exact representations of all consecutive approximations. In this model case, the influence of the algorithm parameters on the method efficiency is analyzed. The approach under study is implemented for solving a problem with a sophisticated configuration of boundaries while using a high precision finite elementmethod to solve the interior boundary value problems. The convergence rate of the iterations and the achieved accuracy of the computations are illustrated with some numerical experiments.     

Nonlocal initial problems for coupled reaction–diffusion systems and their applications

The purpose of this work is to investigate the uniqueness and existence of nonlocal initial problems for a system of nonlinear parabolic equations weakly coupled with ordinary differential equations. The system of equations is considered in bounded and unbounded spatial domains. The uniqueness of the classical solution is proved by means of comparison principles for differential inequalities. The existence of the unique solution is obtained via a monotone iterative method. Applications are given to some model problems in epidemiology and ecology.     

Parameters optimization of selected casting processes using teaching–learning-based optimization algorithm

In the present work, mathematical models of three important casting processes are considered namely squeeze casting, continuous casting and die casting for the parameters optimization of respective processes. A recently developed advanced optimization algorithm named as teaching–learning-based optimization (TLBO) is used for the parameters optimization of these casting processes. Each process is described with a suitable example which involves respective process parameters. The mathematical model related to the squeeze casting is a multi-objective problem whereas the model related to the continuous casting is multi-objective multi-constrained problem and the problem related to the die casting is a single objective problem. The mathematical models which are considered in the present work were previously attempted by genetic algorithm and simulated annealing algorithms. However, attempt is made in the present work to minimize the computational efforts using the TLBO algorithm. Considerable improvements in results are obtained in all the cases and it is believed that a global optimum solution is achieved in the case of die casting process.     

Solving coupled pseudo-parabolic equation using a modified double Laplace decomposition method

In this paper, the modification of double Laplace decomposition method is proposed for the analytical approximation solution of a coupled system of pseudo-parabolic equation with initial conditions. Some examples are given to support our presented method. In addition, we prove the convergence of double Laplace transform decomposition method applied to our problems.     

A GCV based Arnoldi-Tikhonov regularization method

For the solution of linear discrete ill-posed problems, in this paper we consider the Arnoldi-Tikhonov method coupled with the Generalized Cross Validation for the computation of the regularization parameter at each iteration. We study the convergence behavior of the Arnoldi method and its properties for the approximation of the (generalized) singular values, under the hypothesis that Picard condition is satisfied. Numerical experiments on classical test problems and on image restoration are presented.     

无界区域上基于自然边界归化的双调和方程的一种重叠型区域分解法

1．引言双调和方程边值问题的一个力学背景是薄板弯曲问题．对于有界区域上的双调和方程，可以直接利用协调元和非协调元求解18，10］．我们考虑双调和方程Dirichlet外边值问题其中0是充分光滑闭曲线ro之外的无界区域，naro关于0的单位外法向量．引理1．且卜］．若。0EH’／‘（几），gEH‘／‘（fo），则问题（1．1）在W0z（fi）中有唯一解·这里由迹定理，可找到一个具有紧支集的函数识。。，g）E护（炉）满足可。0；g）【F。＝。0和则问题（1．1）等价于如下齐次边值问题其中f—一面‘B有紧支集．边值问题（工．2）又可转化为变分问…     

Preconditioned Dynamic Iteration for Coupled Differential-Algebraic Systems

The network approach to the modelling of complex technical systems results frequently in a set of differential-algebraic systems that are connected by coupling conditions. A common approach to the numerical solution of such coupled problems is based on the coupling of standard time integration methods for the subsystems. As a unified framework for the convergence analysis of such multi-rate, multi-method or dynamic iteration approaches we study in the present paper the convergence of a dynamic iteration method with a (small) finite number of iteration steps in each window. Preconditioning is used to guarantee stability of the coupled numerical methods. The theoretical results are applied to quasilinear problems from electrical circuit simulation and to index-3 systems arising in multibody dynamics.     

On the convergence condition for the Schwarz alternating method for the two-dimensional Laplace equation

The Schwarz alternating method makes it possible to construct a solution of the Dirichlet problem for the two-dimensional Laplace equation in a finite union of overlapping domains, provided that this problem has a solution in each domain. The existing proof of the method convergence and estimation of the convergence rate use the condition that the normals to the boundaries of the domains at the intersection points are different. In the paper, it is proved that this constraint can be removed for domains with Hölder continuous normals. Removing the constraint does not affect the rate of convergence.

     

Coupling Canonical Boundary Element Method with FEM to Solve Harmonic Problem over Cracked Domain

Using the canonical boundary reduction, suggested by Feng Kang, coupled with the finite element method, this paper gives the numerical solutions of the harmonic boundary-value problem over the domain with crack or concave angle. When the coupling is conforming, convergence and error estimates are obtained. This coupling removes the limitation of the canonical boundary reduction to some typical domains, and avoids the shortcoming of the classical finite element method, because of which the accuracy is damaged seriously and the approximate solution does not reflect the behaviour of the solution near the singularity. Numerical calculations have verified those conclusions.     

Time stepping for vectorial operator splitting

We present a fully implicit finite difference method for the unsteady incompressible Navier-Stokes equations. It is based on the one-step θ-method for discretization in time and a special coordinate splitting (called vectorial operator splitting) for efficiently solving the nonlinear stationary problems for the solution at each new time level. The resulting system is solved in a fully coupled approach that does not require a boundary condition for the pressure. A staggered arrangement of velocity and pressure on a structured Cartesian grid combined with the fully implicit treatment of the boundary conditions helps us to preserve the properties of the differential operators and thus leads to excellent stability of the overall algorithm. The convergence properties of the method are confirmed via numerical experiments.     

Domain of existence for the solution of some IVP's and BVP's by using an efficient ninth-order iterative method

In this paper, we consider the problem of solving initial value problems and boundary value problems through the point of view of its continuous form. It is well known that in most cases these types of problems are solved numerically by performing a discretization and applying the finite difference technique to approximate the derivatives, transforming the equation into a finite-dimensional nonlinear system of equations. However, we would like to focus on the continuous problem, and therefore, we try to set the domain of existence and uniqueness for its analytic solution. For this purpose, we study the semilocal convergence of a Newton-type method with frozen first derivative in Banach spaces. We impose only the assumption that the Fréchet derivative satisfies the Lipschitz continuity condition and that it is bounded in the whole domain in order to obtain appropriate recurrence relations so that we may determine the domains of convergence and uniqueness for the solution. Our final aim is to apply these theoretical results to solve applied problems that come from integral equations, ordinary differential equations, and boundary value problems.     

A domain decomposition method using efficient interface-acting preconditioners

The conjugate gradient boundary iteration (CGBI) is a domain decomposition method for symmetric elliptic problems on domains with large aspect ratio. High efficiency is reached by the construction of preconditioners that are acting only on the subdomain interfaces. The theoretical derivation of the method and some numerical results revealing a convergence rate of 0.04-0.1 per iteration step are given in this article. For the solution of the local subdomain problems, both finite element (FE) and spectral Chebyshev methods are considered.

     

Adaptive finite elements for exterior domain problems

Summary. We present an adaptive finite element method for solving elliptic problems in exterior domains, that is for problems in the exterior of a bounded closed domain in , . We describe a procedure to generate a sequence of bounded computational domains , , more precisely, a sequence of successively finer and larger grids, until the desired accuracy of the solution is reached. To this end we prove an a posteriori error estimate for the error on the unbounded domain in the energy norm by means of a residual based error estimator. Furthermore we prove convergence of the adaptive algorithm. Numerical examples show the optimal order of convergence. Received July 8, 1997 /Revised version received October 23, 1997