On Adomian's decomposition method and some comparisons with Picard's iterative scheme

We consider the general mathematical framework of Adomian's decomposition method (G. Adomian, “Stochastic Systems,” Academic Press, New York, 1983) for a large class of nonlinear operator equations. Picard's iterative scheme is considered for the same class for equations and compared with the decomposition method. The paper identifies carefully all substantial differences between the two methods and shows that various advantages exist for the decomposition method.     

Adomian's method of decomposition: critical review and examples of failure

Adomian's method of decomposition is considered in application to initial-boundary value problems for the one space-dimensional spatially homogeneous heat conduction equation. It is shown that the fundamental equation of the method is well-defined only for certain restricted types of boundary conditions. Within the class of such boundary conditions, examples are given such that the fundamental equation fails to have a unique solution, and such that the sequence produced by iteration of this equation is divergent. The latter is a counterexample to a published assertion of convergence.     

Initial/boundary value problems for the semidiscrete Boltzmann equation: Analysis by Adomian's decomposition method

This paper provides a solution technique based on Adomian's decomposition method for a large class of initial/boundary value problems for the semidiscrete Boltzmann equation: a partial differential-integral equation of a semilinear type, in the kinetic theory of gases. The paper also proposes a use of the decomposition method as an algorithm for the continuous approximation of the solutions in a discretized time-space domain.     

Application of He's variational iteration method and Adomian's decomposition method to Sawada–Kotera–Ito seventh‐order equation

In this article, the Sawada–Kotera–Ito seventh‐order equation is studied. He's variational iteration method and Adomian's decomposition method (ADM) are applied to obtain solution of this equation. We compare these methods together. The study highlights the significant features of the employed methods and its capability of handling completely integrable equations. © 2010 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 27: 887–897, 2011     

On the Adomian (decomposition) method and comparisons with Picard's method

This paper shows that the Adomian decomposition method (“Stochastic Systems,” Academic Press, New York,1983) is not to be identified with any of the earlier methods since it solves a wide class of nonlinear and stochastic equations without the often unphysical assumptions which have become customary. Similarities between the Picard method and the decomposition method are purely superficial.     

Error estimate for a stabilised domain decomposition method with nonmatching grids

Summary. In this paper, we analyse a stabilisation technique for the so-called three-field formulation for nonoverlapping domain decomposition methods. The stabilisation is based on boundary bubble functions in each subdomain which are then eliminated by static condensation. The discretisation grids in the subdomains can be chosen independently as well as the grid for the final interface problem. We present the analysis of the method and we construct a set of bubble functions which guarantees the optimal rate of convergence. Received May 12, 1998 / Revised version received November 21, 2000 / Published online June 7, 2001     

Analysis of a stabilized three-fields domain decomposition method

Summary. In this paper we prove that, for suitable choices of the bilinear form involved in the stabilization procedure, the stabilized three fields domain decomposition method proposed in [8] is stable and convergent uniformly in the number of subdomains and with respect to their sizes under quite general assumptions on the decomposition and on the discretization spaces. The same is proven to hold for the resulting discrete Steklov-Poincaré operator. Received April 4, 2000 / Revised version received January 9, 2001 / Published online June 17, 2002     

On the convergence rate of a parallel nonoverlapping domain decomposition method

In recent years,a nonoverlapping domain decomposition iterative procedure,which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces,has been developed and analyzed.It is known that the convergence rate of this method is 1-O(h),where h is mesh size.In this paper,the convergence rate is improved to be 1-O(h1/2 H-1/2)sometime by choosing suitable parameter,where H is the subdomain size.Counter examples are constructed to show that our convergence estimates are sharp,which means that the convergence rate cannot be better than 1-O(h1/2H-1/2)in a certain case no matter how parameter is chosen.     

On a domain decomposition method for the computation of conformal modules

We consider a domain decomposition method for approximating the conformal modules of a certain class of long quadrilaterals.     

Application of Adomian's approximation to blood flow through arteries in the presence of a magnetic field

The present investigation deals with the application of Adomian's decomposition method to blood flow through a constricted artery in the presence of an external transverse magnetic field which is applied uniformly. The blood flowing through the tube is assumed to be Newtonian in character. The expressions for the two-term approximation to the solution of stream function, axial velocity component and wall shear stress are obtained in this analysis. The numerical solutions of the wall shear stress for different values of Reynold number and Hartmann number are shown graphically. The solution of this theoretical result for a particular Hartmann number is compared with the integral method solution of Morgan and Young [17].     

Domain decomposition method for a system of quasivariational inequalities

We propose a domain decomposition method for a system of quasivariational inequalities related to the HJB equation. The monotone convergence of the algorithm is also established.     

Iterative filtering as a direct method for the decomposition of nonstationary signals

Numerical Algorithms - The Iterative Filtering method is a technique developed recently for the decomposition and analysis of nonstationary and nonlinear signals. In this work, we propose two...     

A posteriori error analysis for a boundary element method with nonconforming domain decomposition

We present and analyze an a posteriori error estimator based on mesh refinement for the solution of the hypersingular boundary integral equation governing the Laplacian in three dimensions. The discretization under consideration is a nonconforming domain decomposition method based on the Nitsche technique. Assuming a saturation property, we establish quasireliability and efficiency of the error estimator in comparison with the error in a natural (nonconforming) norm. Numerical experiments with uniform and adaptively refined meshes confirm our theoretical results. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 30: 947–963, 2014     

The Neumann-Dirichlet domain decomposition method with inexact solvers on the subdomains

Summary We consider the Neumann-Dirichlet domain decomposition method for the solution of linear elliptic boundary value problems. We study the following question. Suppose that the auxiliary problems on the subdomains are not solved exactly, but only with a fixed, mesh size independent accuracy. Does the speed of convergence remain mesh size independently bounded? We show that the answer is no in general, but that mesh size independent convergence can be obtained if the accuracy requirement on the subsolvers becomes increasingly severe as the mesh size tends to zero.     

A meshfree method with a non-overlapping domain decomposition method based on TPS for solving the forward-backward heat equation in two-dimension

Numerical Algorithms - We present a truly meshless method based on the thin plate splines for the numerical solution of the two dimensional forward-backward heat equation and give a robust...     

A domain decomposition method for a kind of optimization problems

In this paper, we are concerned with the monotone convergence of a multiplicative method for solving a kind of optimization problems. We show that the iterate sequence produced by the method converges to the solution of the problem monotonically.     

Exponential attractors for a class of evolution equations by a decomposition method

Our aim in this Note is to prove the existence of exponential attractors for a class of evolution equations that includes some models of generalized Cahn-Hilliard equations described in [5] by M. Gurtin. Our proof is based on a decomposition of the difference of two solutions similar to that introduced in [1] by A. Babin and B. Nicolaenko.     

An analysis of the BEM-FEM non-overlapping domain decomposition method for a scattering problem

In this paper, we analyze the BEM-FEM non-overlapping domain decomposition method introduced in Boubendir [Techniques de Décomposition de Domaine et Méthode d’Equations Intégrales, Ph.D. Thesis, INSA, Toulouse, France, 2002] and improved in Boubendir et al. [A coupling BEM-FEM method using a FETI-LIKE domain decomposition method, in: Proceedings of Waves 2005, Providence, RI, 2005, pp. 188–190] and Bendali et al. [A FETI-like domain decomposition method for coupling FEM and BEM in large-size problems of acoustic scattering, to appear.] The transmission conditions used in this method introduce a quantity that prevents the approach of Després [Méthodes de décomposition de domaine pour les problèmes de propagation d’ondes en régime harmonique, Le théorème de Borg pour l’équation de Hill vectorielle, Ph.D. Thesis, Paris VI University, France, 1991] to establish convergence to be adapted. However, we show that convergence can be established when the geometry allows for a decomposition of the solution into propagating and evanescent portions with a methodology based on modal analysis. Here, we exemplify this in the case of circular cylindrical geometries where the derivations can be based on properties of Bessel functions.     

On the domain decomposition method for the generalized Stokes problem with continuous pressure

Using the nonoverlapping domain decomposition approach, we propose a formulation of the dual Schur algorithm for the generalized Stokes problem discretized by a mixed finite element method continuous for the pressure in each subdomain, but discontinuous at the interfaces. The corresponding LBB condition is checked. The dual interface problem is written in the case of two subdomains, and it is generalized to several subdomains. An efficient preconditioner for the interface problem is derived. Numerical results are presented for two different local solvers. Parallel computations were made on an IBM‐SP2. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 84–106, 2000