Uniform convergence of Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity

In this article we prove uniform convergence estimates for the recently developed Galerkin‐multigrid methods for nonconforming finite elements for second‐order problems with less than full elliptic regularity. These multigrid methods are defined in terms of the “Galerkin approach,” where quadratic forms over coarse grids are constructed using the quadratic form on the finest grid and iterated coarse‐to‐fine intergrid transfer operators. Previously, uniform estimates were obtained for problems with full elliptic regularity, whereas these estimates are derived with less than full elliptic regularity here. Applications to the nonconforming P₁, rotated Q₁, and Wilson finite elements are analyzed. The result applies to the mixed method based on finite elements that are equivalent to these nonconforming elements. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 203–217, 2002; DOI 10.1002/num.10004     

New initial-value method for singularly perturbed boundary-value problems

An initial-value method is given for second-order singularly perturbed boundary-value problems with a boundary layer at one endpoint. The idea is to replace the original two-point boundary value problem by two suitable initial-value problems. The method is very easy to use and to implement. Nontrivial text problems are used to show the feasibility of the given method, its versatility, and its performance in solving linear and nonlinear singularly perturbed problems.This work was supported in part by the Consiglio Nazionale delle Ricerche, Contract No. 86.02108.01, and in part by the Ministero della Pubblica Istruzione.     

Demonstration of Soliton Solutions of DNLS Equation by Liouville Theorem

In the inverse scattering transform (IST), the reflectionless Jost solutions are combined by their analytic properties in the complex spectrum parameter plane, and then can be shown to satisfy the two Lax equations indeed by Liouville theorem. So the corresponding soliton solutions certainly satisfy the nonlinear equation by compatibility condition. Especially the multi-soliton solutions of DNLS equation can be demonstrated in this way. PACS Numbers: 05.45.Yv, 02.30.-f, 11.10.Ef     

Flow in a waterfall with large Froude number

An approximate method is developed to solve the full nonlinear equations governing two-dimensional irrotational flow in a free waterfall, falling under the influence of gravity, at high Froude number based on conditions far upstream. Schwarz—Christoffel transformation is used to map the region, in the complex potential-plane, onto the upper half-plane. The Hilbert transformation as well as the perturbation technique, for large Froude number, are used as a basis for the approximate solution of the problem. A complete solution, up to second-order approximation, for the downstream free-surfaces profiles, for different Froude number, is discussed and illustrated. The obtained approximate solutions are compared with those of other authors. Favourable agreement with other results suggests that this method is effective in dealing with flow problems strongly influenced by gravity and high Froude number. The results obtained by this method are sufficiently accurate for practical purposes.     

Resolvent estimates in W−1,p related to strongly coupled‐linear parabolic systems with coupled nonsmooth capacities

We investigate linear parabolic systems with coupled nonsmooth capacities and mixed boundary conditions. We prove generalized resolvent estimates in W^{?1, p} spaces. The method is an appropriate modification of a technique introduced by Agmon to obtain L^p estimates for resolvents of elliptic differential operators in the case of smooth boundary conditions. Moreover, we establish an existence and uniqueness result. Copyright © 2007 John Wiley & Sons, Ltd.     

Carathéodory's method for a class of second order differential equations on the half-line

The method of equivalent variational methods, originally due to Carathéodory for free problems in the calculus of variations is extended to investigate boundary value problems for a class of second order differential equations on the half-line. Some applications are presented to illustrate the potential of this method.     

Time Asymptotic Behaviour for a One-Velocity Transport Operator with Maxwell Boundary Condition

This paper is concerned with the spectral analysis of a one-velocity transport operator with Maxwell boundary condition in L ₁-space. After a detailed spectral analysis it is shown that the associated Cauchy problem is governed by a C ₀-semigroup. Next, we discuss the irreducibility of the transport semigroup. In particular, we show that the transport semigroup is irreducible. Finally, a spectral decomposition of the solutions into an asymptotic term and a transient one which will be estimated for smooth initial data is given.     

A mathematical model of the ring-spinning process

This paper develops a mathematical model of the ring-spinning process that takes into account its non-stationary nature. A complex system of differential equations is obtained, which from a mathematical point of view constitutes a ‘free-boundary’ problem. Its solution involves definition of suitable boundary conditions related to the mechanical characteristics of the process and of the spinning machine itself. The boundary conditions which determine the solution are pointed out. A numerical solution of the system of differential equations can be obtained by the Finite-Segments method, as shown in an example.     

一类非对称椭圆问题瀑布型多重网格法

本将瀑布型多重网格法用于求解非对称椭圆边值问题，数值结果表明算法是有效的。     

A note on stability of the Douglas splitting method

In this note some stability results are derived for the Douglas splitting method. The relevance of the theoretical results is tested for an advection-reaction equation.