Variational Poisson-Nijenhuis structures for partial differential equations

We explore variational Poisson-Nijenhuis structures on nonlinear partial differential equations and establish relations between the Schouten and Nijenhuis brackets on the initial equation and the Lie bracket of symmetries on its natural extensions (coverings). This approach allows constructing a framework for the theory of nonlocal structures. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 154, No. 2, pp. 268–282, February, 2008.     

Part IV Parallel algorithms for partial differential equations

Part IV Parallel algorithms for partial differential equations     

Analysis of a moving collocation method for one-dimensional partial differential equations

A moving collocation method has been shown to be very efficient for the adaptive solution of second- and fourth-order time-dependent partial differential equations and forms the basis for the two robust codes MOVCOL and MOVCOL4.In this paper,the relations between the method and the traditional collocation and finite volume methods are investigated.It is shown that the moving collocation method inherits desirable properties of both methods: the ease of implementation and high-order convergence of the traditional collocation method and the mass conservation of the finite volume method.Convergence of the method in the maximum norm is proven for general linear two-point boundary value problems.Numerical results are given to demonstrate the convergence order of the method.     

散度形式椭圆型方程弱下解的局部估计

运用De Giorgi迭代技巧,得到了一般散度形式的椭圆型偏微分方程弱下解的局部估计,推广了有关文献中的结论.     

Local solvability for semilinear anisotropic partial differential equations

In this paper we introduce a class of Gevrey-Sobolev anisotropic spaces and, by means of the application of the Banach fixed point Theorem, we obtain the solvability of a related semilinear equation. Entrata in Redazione il 21 luglio 1999.     

Local elimination algorithms for solving sparse discrete problems

The class of local elimination algorithms is considered that make it possible to obtain global information about solutions of a problem using local information. The general structure of local elimination algorithms is described that use neighborhoods of elements and the structural graph describing the problem structure; an elimination algorithm is also described. This class of algorithms includes local decomposition algorithms for discrete optimization problems, nonserial dynamic programming algorithms, bucket elimination algorithms, and tree decomposition algorithms. It is shown that local elimination algorithms can be used for solving optimization problems.     

Local solvability and hypoellipticity for semilinear anisotropic partial differential equations

We propose a unified approach, based on methods from microlocal analysis, for characterizing the local solvability and hypoellipticity in and Gevrey classes of -variable semilinear anisotropic partial differential operators with multiple characteristics. The conditions imposed on the lower-order terms of the linear part of the operator are optimal.

     

Preconditioning discretizations of systems of partial differential equations

This survey paper is based on three talks given by the second author at the London Mathematical Society Durham Symposium on Computational Linear Algebra for Partial Differential Equations in the summer of 2008. The main focus will be on an abstract approach to the construction of preconditioners for symmetric linear systems in a Hilbert space setting. Typical examples that are covered by this theory are systems of partial differential equations which correspond to saddle point problems. We will argue that the mapping properties of the coefficient operators suggest that block diagonal preconditioners are natural choices for these systems. To illustrate our approach a number of examples will be considered. In particular, parameter‐dependent systems arising in areas like incompressible flow, linear elasticity, and optimal control theory will be studied. The paper contains analysis of several models which have previously been discussed in the literature. However, here each example is discussed with reference to a more unified abstract approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Finite integration method for partial differential equations

A finite integration method is proposed in this paper to deal with partial differential equations in which the finite integration matrices of the first order are constructed by using both standard integral algorithm and radial basis functions interpolation respectively. These matrices of first order can directly be used to obtain finite integration matrices of higher order. Combining with the Laplace transform technique, the finite integration method is extended to solve time dependent partial differential equations. The accuracy of both the finite integration method and finite difference method are demonstrated with several examples. It has been observed that the finite integration method using either radial basis function or simple linear approximation gives a much higher degree of accuracy than the traditional finite difference method.     

Kinetic decomposition for singularly perturbed higher order partial differential equations

In this paper, we consider singularly perturbed higher order partial differential equations. We establish the condition under which the approximate solutions converge in a strong topology to the entropy solution of a scalar conservation laws using methodology developed in Hwang and Tzavaras (Comm. Partial Differential Equations 27 (2002) 1229). First, we obtain the approximate transport equation for the given dispersive equations. Then using the averaging lemma, we obtain the convergence.     

Local Linearization method for the numerical solution of stochastic differential equations

The Local Linearization (LL) approach for the numerical solution of stochastic differential equations (SDEs) is extended to general scalar SDEs, as well as to non-autonomous multidimensional SDEs with additive noise. In case of autonomous SDEs, the derivation of the method introduced gives theoretical support to one of the previously proposed variants of the LL approach. Some numerical examples are given to demonstrate the practical performance of the method.     

Two-dimensional state-space discrete models for hyperbolic partial differential equations

It is shown that the finite difference technique can be used to transform some important linear distributed processes described by partial differential equations into so-called Roesser discrete state-space models. Two illustrative examples of numerical solutions are given and compared with exact solutions.     

Nonstationary monotone iterative methods for nonlinear partial differential equations

A simple technique is given in this paper for the construction and analysis of monotone iterative methods for a class of nonlinear partial differential equations. With the help of the special nonlinear property we can construct nonstationary parameters which can speed up the iterative process in solving the nonlinear system. Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive solutions. The adaptive meshes are generated by the 1-irregular mesh refinement scheme which together with the M-matrix of the finite element stiffness matrix lead to existence–uniqueness–comparison theorems with simple upper and lower solutions as initial iterates. Some numerical examples, including a test problem with known analytical solution, are presented to demonstrate the accuracy and efficiency of the adaptive and monotone properties. Numerical results of simulations on a MOSFET with the gate length down to 34 nm are also given.     

Discontinuous Legendre wavelet element method for elliptic partial differential equations

By incorporating the Legendre multiwavelet into the discontinuous Galerkin (DG) method, this paper presents a novel approach for solving Poisson’s equation with Dirichlet boundary, which is known as the discontinuous Legendre multiwavelet element (DLWE) method, derive an adaptive algorithm for the method, and estimate the approximating error of its numerical fluxes. One striking advantage of our method is that the differential operator, boundary conditions and numerical fluxes involved in the elementwise computation can be done with lower time cost. Numerical experiments demonstrate the validity of this method. Furthermore, this paper generalizes the DLWE method to the general elliptic equations defined on a bounded domain and describes the possibilities of constructing optimal adaptive algorithm. The proposed method and its generalizations are also applicable to some other kinds of partial differential equations.     

Numerical methods for nonlinear partial differential equations of fractional order

In this article, we implement relatively new analytical techniques, the variational iteration method and the Adomian decomposition method, for solving nonlinear partial differential equations of fractional order. The fractional derivatives are described in the Caputo sense. The two methods in applied mathematics can be used as alternative methods for obtaining analytic and approximate solutions for different types of fractional differential equations. In these schemes, the solution takes the form of a convergent series with easily computable components. Numerical results show that the two approaches are easy to implement and accurate when applied to partial differential equations of fractional order.     

A brief review of some application driven fast algorithms for elliptic partial differential equations

Some application driven fast algorithms developed by the author and his collaborators for elliptic partial differential equations are briefly reviewed here. Subsequent use of the ideas behind development of these algorithms for further development of other algorithms some of which are currently in progress is briefly mentioned. Serial and parallel implementation of these algorithms and their applications to some pure and applied problems are also briefly reviewed.     

Preconditioned Barzilai-Borwein method for the numerical solution of partial differential equations

The preconditioned Barzilai-Borwein method is derived and applied to the numerical solution of large, sparse, symmetric and positive definite linear systems that arise in the discretization of partial differential equations. A set of well-known preconditioning techniques are combined with this new method to take advantage of the special features of the Barzilai-Borwein method. Numerical results on some elliptic test problems are presented. These results indicate that the preconditioned Barzilai-Borwein method is competitive and sometimes preferable to the preconditioned conjugate gradient method.This author was partially supported by the Parallel and Distributed Computing Center at UCV.This author was partially supported by BID-CONICIT, project M-51940.