A nonlinear iteration method for solving a two-dimensional nonlinear coupled system of parabolic and hyperbolic equations

A nonlinear iteration method for solving a class of two-dimensional nonlinear coupled systems of parabolic and hyperbolic equations is studied. A simple iterative finite difference scheme is designed; the calculation complexity is reduced by decoupling the nonlinear system, and the precision is assured by timely evaluation updating. A strict theoretical analysis is carried out as regards the convergence and approximation properties of the iterative scheme, and the related stability and approximation properties of the nonlinear fully implicit finite difference (FIFD) scheme. The iterative algorithm has a linear constringent ratio; its solution gives a second-order spatial approximation and first-order temporal approximation to the real solution. The corresponding nonlinear FIFD scheme is stable and gives the same order of approximation. Numerical tests verify the results of the theoretical analysis. The discrete functional analysis and inductive hypothesis reasoning techniques used in this paper are helpful for overcoming difficulties arising from the nonlinearity and coupling and lead to a related theoretical analysis for nonlinear FI schemes.     

Error Estimates for Multilevel Approximation Using Polyharmonic Splines

Polyharmonic splines are used to interpolate data in a stationary multilevel iterative refinement scheme. By using such functions the necessary tools are provided to obtain simple pointwise error bounds on the approximation. Linear convergence between levels is shown for regular data on a scaled multiinteger grid, and a multilevel domain decomposition method.     

一类非线性系统的分段迭代学习控制

本文提出并研究一类非线性系统的分段迭代学习控制问题.基于P型学习律和D型学习律构建得到分段迭代学习控制律,利用压缩映射原理,证明这种分段迭代学习律能使得系统的输出跟踪误差沿迭代轴方向收敛.仿真算例验证了算法的有效性.     

A skew-symmetric iterative method for solving a steady convection–diffusion–reaction equation with an indefinite reaction coefficient

An iterative product-type triangular skew-symmetric method (PTSM) is used to solve systems of linear algebraic equations (SLAEs) obtained by approximation with a central-difference scheme of a first-type boundary value problem for convection–diffusion–reaction and standard grid ordering. Sufficient conditions for non-negative definiteness of the SLAE matrix resulting from this approximation are obtained for the indefinite reaction coefficient. This property provides convergence of a wide class of iterative methods (in particular, the PTSM). In test problems, agreement of the theory with computational experiments is shown, and a comparison of the PTSM and SSOR is done.     

Integral boundary value problem for impulsive integro-differential equations in Banach spaces

In this paper, by using the cone theory and monotone iterative technique, we investigate the existence of extremal solutions and unique solution of the integral boundary value problem for a class of first-order impulsive integro-differential equations in a real Banach space. An explicit iterative scheme for the unique solution and an error estimate of the approximation sequence are also derived.     

Numerical performance of parallel group explicit solvers for the solution of fourth order elliptic equations

Many applications in applied mathematics and engineering involve numerical solutions of partial differential equations (PDEs). Various discretisation procedures such as the finite difference method result in a problem of solving large, sparse systems of linear equations. In this paper, a group iterative numerical scheme based on the rotated (skewed) five-point finite difference discretisation is proposed for the solution of a fourth order elliptic PDE which represents physical situations in fluid mechanics and elasticity. The rotated approximation formulas lead to schemes with lower computational complexities compared to the centred approximation formulas since the iterative procedure need only involve nodes on half of the total grid points in the solution domain. We describe the development of the parallel group iterative scheme on a cluster of distributed memory parallel computer using Message-Passing Interface (MPI) programming environment. A comparative study with another group iterative scheme derived from the centred difference formula is also presented. A detailed performance analysis of the parallel implementations of both group methods will be reported and discussed.     

Analysis of a new BEM‐FEM coupling for two‐dimensional fluid‐solid interaction

In this work we present a new numerical method, based on a coupling of finite and boundary elements, to solve a fluid‐solid interaction problem in the plane. The discrete method uses classical Lagrange finite elements adapted to curved boundaries for the field variable and spectral approximation of the unknowns on the artificial boundary. We provide error estimates for this Galerkin scheme and propose a full discretization based on elementary quadrature formulae, showing that the perturbation due to numerical integration preserves the optimal rate of convergence. We also suggest an iterative method to solve the complicated linear systems arising from this type of schemes. © 2005 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005     

Convergence analysis of a projection semi-implicit coupling scheme for fluid–structure interaction problems

In this paper, we provide a convergence analysis of a projection semi-implicit scheme for the simulation of fluid–structure systems involving an incompressible viscous fluid. The error analysis is performed on a fully discretized linear coupled problem: a finite element approximation and a semi-implicit time-stepping strategy are respectively used for space and time discretization. The fluid is described by the Stokes equations, the structure by the classical linear elastodynamics equations (linearized elasticity, plate or shell models) and all changes of geometry are neglected. We derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least ${\sqrt{\delta t}}In this paper, we provide a convergence analysis of a projection semi-implicit scheme for the simulation of fluid–structure systems involving an incompressible viscous fluid. The error analysis is performed on a fully discretized linear coupled problem: a finite element approximation and a semi-implicit time-stepping strategy are respectively used for space and time discretization. The fluid is described by the Stokes equations, the structure by the classical linear elastodynamics equations (linearized elasticity, plate or shell models) and all changes of geometry are neglected. We derive an error estimate in finite time and we prove that the time discretization error for the coupling scheme is at least ?{dt}{\sqrt{\delta t}}. Finally, some numerical experiments that confirm the theoretical analysis are presented.     

Approximating zero-variance importance sampling in a reliability setting

We consider a class of Markov chain models that includes the highly reliable Markovian systems (HRMS) often used to represent the evolution of multicomponent systems in reliability settings. We are interested in the design of efficient importance sampling (IS) schemes to estimate the reliability of such systems by simulation. For these models, there is in fact a zero-variance IS scheme that can be written exactly in terms of a value function that gives the expected cost-to-go (the exact reliability, in our case) from any state of the chain. This IS scheme is impractical to implement exactly, but it can be approximated by approximating this value function. We examine how this can be effectively used to estimate the reliability of a highly-reliable multicomponent system with Markovian behavior. In our implementation, we start with a simple crude approximation of the value function, we use it in a first-order IS scheme to obtain a better approximation at a few selected states, then we interpolate in between and use this interpolation in our final (second-order) IS scheme. In numerical illustrations, our approach outperforms the popular IS heuristics previously proposed for this class of problems. We also perform an asymptotic analysis in which the HRMS model is parameterized in a standard way by a rarity parameter ε, so that the relative error (or relative variance) of the crude Monte Carlo estimator is unbounded when ε→0. We show that with our approximation, the IS estimator has bounded relative error (BRE) under very mild conditions, and vanishing relative error (VRE), which means that the relative error converges to 0 when ε→0, under slightly stronger conditions.     

A convergence analysis for a sweeping preconditioner for block tridiagonal systems of linear equations

We study sweeping preconditioners for symmetric and positive definite block tridiagonal systems of linear equations. The algorithm provides an approximate inverse that can be used directly or in a preconditioned iterative scheme. These algorithms are based on replacing the Schur complements appearing in a block Gaussian elimination direct solve by hierarchical matrix approximations with reduced off‐diagonal ranks. This involves developing low rank hierarchical approximations to inverses. We first provide a convergence analysis for the algorithm for reduced rank hierarchical inverse approximation. These results are then used to prove convergence and preconditioning estimates for the resulting sweeping preconditioner. Copyright © 2014 John Wiley & Sons, Ltd.     

Defect-correction multigrid methods for nonlinear problems

Auzinger and Stetter [1] combine a multigrid method with defect-correction iteration and derive a composite iterative procedure which they call the DCMG (defect-correction multigrid) cycle. Using a high-order discrete operator in the coarsegrid correction and a lower-order operator in relaxation, the DCMG cycle achieves the higher-order approximation [4]. In an analogous way, DCMG can be used to solve nonlinear PDEs by using the nonlinear operator in correction and a related linear operator in relaxation. We prove convergence of such a DCMG scheme and give an error estimation.     

On well-posedness of an initial value problem for nonlinear second-order impulsive integro-differential equations of Volterra type in Banach spaces

In this paper, we develop an existence and uniqueness theorem of the solution to an initial value problems for a class of second-order impulsive integro-differential equations of Volterra type in a real Banach space by using the generalized Banach fixed point theorem. An explicit iterative scheme for the solution and an error estimate of the approximation sequence for the initial value problem are also derived. Two examples are presented to demonstrate the application of our results.     

A Dirichlet–Neumann reduced basis method for homogeneous domain decomposition problems

Reduced basis methods allow efficient model reduction of parametrized partial differential equations. In the current paper, we consider a reduced basis method based on an iterative Dirichlet–Neumann coupling for homogeneous domain decomposition of elliptic PDEʼs. We gain very small basis sizes by an efficient treatment of problems with a-priori known geometry. Moreover iterative schemes may offer advantages over other approaches in the context of parallelization. We prove convergence of the iterative reduced scheme, derive rigorous a-posteriori error bounds and provide a full offline/online decomposition. Different methods for basis generation are investigated, in particular a variant of the POD-Greedy procedure. Experiments confirm the rigor of the error estimators and identify beneficial basis construction procedures.     

Weighted iterative solutions of linear differential equations and heat conduction of ideal gases in moment theory

An iterative method is proposed to find a particular solution of a system of linear differential equations, in the form of a fixed-point problem, with no boundary conditions. To circumvent the unboundedness of differential operators, iterative approximation with gradually decreasing weight is used. Conditions for convergence that can easily be checked in numerical iterations are established. Furthermore, for the numerical iterative scheme, uniqueness and stability theorems are proved. These results are applied to heat conduction of ideal gases in moment theory.     

Modified Defect Correction Algorithms for ODEs. Part I: General Theory

The well-known method of Iterated Defect Correction (IDeC) is based on the following idea: Compute a simple, basic approximation and form its defect w.r.t. the given ODE via a piecewise interpolant. This defect is used to define an auxiliary, neighboring problem whose exact solution is known. Solving the neighboring problem with the basic discretization scheme yields a global error estimate. This can be used to construct an improved approximation, and the procedure can be iterated. The fixed point of such an iterative process corresponds to a certain collocation solution. We present a variety of modifications to this algorithm. Some of these have been proposed only recently, and together they form a family of iterative techniques, each with its particular advantages. These modifications are based on techniques like defect quadrature (IQDeC), defect interpolation (IPDeC), and combinations thereof. We investigate the convergence on locally equidistant and nonequidistant grids and show how superconvergent approximations can be obtained. Numerical examples illustrate our considerations. The application to stiff initial value problems will be discussed in Part II of this paper.     

Weighted iterative solutions of linear differential equations and heat conduction of ideal gases in moment theory

An iterative method is proposed to find a particular solution of a system of linear differential equations, in the form of a fixed-point problem, with no boundary conditions. To circumvent the unboundedness of differential operators, iterative approximation with gradually decreasing weight is used. Conditions for convergence that can easily be checked in numerical iterations are established. Furthermore, for the numerical iterative scheme, uniqueness and stability theorems are proved. These results are applied to heat conduction of ideal gases in moment theory.     

A finite difference scheme for nonlinear ultra-parabolic equations

In this paper, our aim is to study a numerical method for an ultraparabolic equation with nonlinear source function. Mathematically, the bibliography on initial–boundary value problems for ultraparabolic equations is not extensive although the problems have many applications related to option pricing, multi-parameter Brownian motion, population dynamics and so forth. In this work, we present the approximate solution by virtue of finite difference scheme and Fourier series. For the nonlinear case, we use an iterative scheme by linear approximation to get the approximate solution and obtain error estimates. A numerical example is given to justify the theoretical analysis.     

Analysis and convergence of the MAC scheme. II. Navier-Stokes equations

The MAC discretization scheme for the incompressible Navier-Stokes equations is interpreted as a covolume approximation to the equations. Using some results from earlier papers dealing with covolume error estimates for div-curl equation systems, and under certain conditions on the data and the solutions of the Navier-Stokes equations, we obtain first-order error estimates for both the vorticity and the pressure.

     

Finite element approximation of an Allen-Cahn/Cahn-Hilliard system

We consider an Allen–Cahn/Cahn–Hilliard system witha non-degenerate mobility and (i) a logarithmic free energyand (ii) a non-smooth free energy (the deep quench limit). Thissystem arises in the modelling of phase separation and orderingin binary alloys. In particular we prove in each case that thereexists a unique solution for sufficiently smooth initial data.Further, we prove an error bound for a fully practical piecewiselinear finite element approximation of (i) and (ii) in one andtwo space dimensions (and three space dimensions for constantmobility). The error bound being optimal in the deep quenchlimit. In addition an iterative scheme for solving the resultingnonlinear discrete system is analysed. Finally some numericalexperiments are presented.     

A convergence analysis result for constrained convex minimization problem

Abstract

Our purpose in this paper is to obtain strong convergence result for approximation of solution to constrained convex minimization problem using a new iterative scheme in a real Hilbert space. Furthermore, we give numerical analysis of our iterative scheme.