Estimates of the first Neumann eigenvalue and the log‐Sobolev constant on non‐convex manifolds

In this paper a number of explicit lower bounds are presented for the first Neumann eigenvalue on non‐convex manifolds. The main idea to derive these estimates is to make a conformal change of the metric such that the manifold is convex under the new metric, which enables one to apply known results obtained in the convex case. This method also works for more general functional inequalities. In particular, some explicit lower bounds are presented for the log‐Sobolev constant on non‐convex manifolds. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Incremental displacement-correction schemes for incompressible fluid-structure interaction

In this paper we introduce a class of incremental displacement-correction schemes for the explicit coupling of a thin-structure with an incompressible fluid. These methods enforce a specific Robin–Neumann explicit treatment of the interface coupling. We provide a general stability and convergence analysis that covers both the incremental and the non-incremental variants. Their stability properties are independent of the added-mass effect. The superior accuracy of the incremental schemes (with respect to the original non-incremental variant) is highlighted by the error estimates, and then confirmed in a benchmark by numerical experiments.     

The C 1 wavelet method for numerical solutions of the Neumann problem

In this article we use the C ¹ wavelet bases on Powell-Sabin triangulations to approximate the solution of the Neumann problem for partial differential equations. The C ¹ wavelet bases are stable and have explicit expressions on a three-direction mesh. Consequently, we can approximate the solution of the Neumann problem accurately and stably. The convergence and error estimates of the numerical solutions are given. The computational results of a numerical example show that our wavelet method is well suitable to the Neumann boundary problem.     

A NEW PERTURBATION THEOREM FOR MOORE-PENROSE METRIC GENERALIZED INVERSE OF BOUNDED LINEAR OPERATORS IN BANACH SPACES

In this paper,we investigate a new perturbation theorem for the Moore-Penrose metric generalized inverses of a bounded linear operator in Banach space. The main tool in this paper is "the generalized Neumann lemma" which is quite different from the method in [12] where "the generalized Banach lemma" was used. By the method of the perturbation analysis of bounded linear operators,we obtain an explicit perturbation theorem and three inequalities about error estimates for the Moore-Penrose metric generalized inverse of bounded linear operator under the generalized Neumann lemma and the concept of stable perturbations in Banach spaces.     

Balancing Neumann–Neumann methods for the cardiac Bidomain model

Balancing Neumann–Neumann preconditioners are constructed, analyzed and numerically studied for the cardiac Bidomain model in three-dimensions. This reaction–diffusion system is discretized by low-order finite elements in space and implicit–explicit methods in time, yielding very ill-conditioned linear systems that must be solved at each time step. The proposed algorithm is based on decomposing the domain into nonoverlapping subdomains and on solving iteratively the Bidomain Schur complement obtained by implicitly eliminating the degrees of freedom interior to each subdomain. The iteration is preconditioned by a Balancing Neumann–Neumann method employing local Neumann solves on each subdomain and a coarse Bidomain solve. A novel approach for the estimation of the average operator of the nonoverlapping decomposition provides a framework for designing coarse spaces for Balancing Neumann–Neumann methods. The theoretical estimates obtained show that the proposed method is scalable, quasi-optimal and robust with respect to possible coefficient discontinuities of the Bidomain operator. The results of extensive parallel numerical tests in three dimensions confirm the convergence rates predicted by the theory.     

Lower bounds of principal eigenvalue in dimension one

For the principal eigenvalue with bilateral Dirichlet boundary condition, the so-called basic estimates were originally obtained by capacitary method. The Neumann case (i.e., the ergodic case) is even harder, and was deduced from the Dirichlet one plus a use of duality and the coupling method. In this paper, an alternative and more direct proof for the basic estimates is presented. The estimates in the Dirichlet case are then improved by a typical application of a recent variational formula. As a dual of the Dirichlet case, the refine problem for bilateral Neumann boundary condition is also treated. The paper starts with the continuous case (one-dimensional diffusions) and ends at the discrete one (birth-death processes). Possible generalization of the results studied here is discussed at the end of the paper.     

Error analysis of the DtN-FEM for the scattering problem in acoustics via Fourier analysis

In this paper, we are concerned with the error analysis for the finite element solution of the two-dimensional exterior Neumann boundary value problem in acoustics. In particular, we establish explicit priori error estimates in H¹ and L²- norms including both the effect of the truncation of the DtN mapping and that of the numerical discretization. To apply the finite element method (FEM) to the exterior problem, the original boundary value problem is reduced to an equivalent nonlocal boundary value problem via a Dirichlet-to-Neumann (DtN) mapping represented in terms of the Fourier expansion series. We discuss essential features of the corresponding variational equation and its modification due to the truncation of the DtN mapping in appropriate function spaces. Numerical tests are presented to validate our theoretical results.     

Dirichlet and Neumann problems for Poisson equation in half lens

In this article, the Green and Neumann functions are given for a half lens and the Dirichlet and Neumann problems for Poisson equation are solved. All formulas are given in explicit form.     

Booster Method for Singularly-Perturbed One-Dimensional Reaction-Diffusion Neumann Problems

A numerical method for singularly-perturbed self-adjoint boundary-value problems for second-order ordinary differential equations subject to Neumann boundary conditions is proposed. In this method (booster method), an asymptotic approximation is incorporated into a finite-difference scheme to improve the numerical solution. Uniform error estimates are derived for this method when implemented in known difference schemes. Numerical examples are presented to illustrate the present method.     

On the principal eigenvalue of a Robin problem with a large parameter

We study the asymptotic behaviour of the principal eigenvalue of a Robin (or generalised Neumann) problem with a large parameter in the boundary condition for the Laplacian in a piecewise smooth domain. We show that the leading asymptotic term depends only on the singularities of the boundary of the domain, and give either explicit expressions or two‐sided estimates for this term in a variety of situations. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

A Thomson’s Principle and a Rayleigh’s Monotonicity Law for Nonlinear Networks

We give a representation of the solution of the Neumann problem for the Laplace operator on the n-dimensional unit ball in terms of the solution of an associated Dirichlet problem. The representation is extended to other operators besides the Laplacian, to smooth simply connected planar domains, and to the infinite-dimensional Laplacian on the unit ball of an abstract Wiener space, providing in particular an explicit solution for the Neumann problem in this case. As an application, we derive an explicit formula for the Dirichlet-to-Neumann operator, which may be of independent interest.     

导子的范数估计

本文给出了因子 ｖｏｎ Ｎｅｕｍａｎｎ代数中套子代数上导子的范数估计．利用此结果得到一个距离公式,并证明了因子 ｖｏｎ Ｎｅｕｍａｎｎ代数中的任何套子代数都具有ＡＩＰ（ｒ, ｓ）性质     

Boundary integral equations on the sphere with radial basis functions: error analysis

Radial basis functions are used to define approximate solutions to boundary integral equations on the unit sphere. These equations arise from the integral reformulation of the Laplace equation in the exterior of the sphere, with given Dirichlet or Neumann data, and a vanishing condition at infinity. Error estimates are proved. Numerical results supporting the theoretical results are presented.     

Lipschitz Estimates in Almost‐Periodic Homogenization

We establish uniform Lipschitz estimates for second‐order elliptic systems in divergence form with rapidly oscillating, almost‐periodic coefficients. We give interior estimates as well as estimates up to the boundary in bounded C^1,α domains with either Dirichlet or Neumann data. The main results extend those in the periodic setting due to Avellaneda and Lin for interior and Dirichlet boundary estimates and later Kenig, Lin, and Shen for the Neumann boundary conditions. In contrast to these papers, our arguments are constructive (and thus the constants are in principle computable) and the results for the Neumann conditions are new even in the periodic setting, since we can treat nonsymmetric coefficients. We also obtain uniform W^1,p estimates.© 2016 Wiley Periodicals, Inc.     

Neumann and Robin problems in a cracked domain with jump conditions on cracks

The boundary value problem for the Laplace equation is studied on a domain with smooth compact boundary and with smooth internal cracks. The Neumann or the Robin condition is given on the boundary of the domain. The jump of the function and the jump of its normal derivative is prescribed on the cracks. The solution is looked for in the form of the sum of a single layer potential and a double layer potential. The solvability of the corresponding integral equation is determined and the explicit solution of this equation is given in the form of the Neumann series. Estimates for the absolute value of the solution of the boundary value problem and for the absolute value of the gradient of the solution are presented.     

Lp-Estimates for a Solution to the Dirichlet Problem and to the Neumann Problem for the Heat Equation in a Wedge with Edge of Arbitrary Codimension

The Dirichlet problem and the Neumann problem in a wedge with edge of an arbitrary codimension are studied. On the basis of the Green functions of these problems in a cone, estimates for solutions are obtained. Coercive estimates for the solutions are also obtained in the Kondrat'ev spaces. Bibliography: 14 titles.     

NATURAL BOUNDARY ELEMENT METHOD FOR THREE DIMENSIONAL EXTERIOR HARMONIC PROBLEM WITH AN INNER PROLATE SPHEROID BOUNDARY

In this paper, we study natural boundary reduction for Laplace equation with Dirichletor Neumann boundary condition in a three-dimensional unbounded domain, which is theoutside domain of a prolate spheroid. We express the Poisson integral formula and naturalintegral operator in a series form explicitly. Thus the original problem is reduced to aboundary integral equation on a prolate spheroid. The variational formula for the reducedproblem and its well-posedness are discussed. Boundary element approximation for thevariational problem and its error estimates, which have relation to the mesh size andthe terms after the series is truncated, are also presented. Two numerical examples arepresented to demonstrate the effectiveness and error estimates of this method.     

Probabilistic approach to the Neumann problem

In this paper, we consider the Neumann boundary value problem of Schrödinger operator with measure potential . First, a martingale formulation of the Neumann problem and an analytic characterization of the martingale formulation are given. Then, by using the Dirichlet forms and Stochastic analysis we obtain an explicit formula for the unique weak solution of this problem in terms of reflecting Brownias motion and it's boundary local time.     

二维区域上椭圆均匀化问题的收敛率（英文）

In this paper, we study the convergence rates of solutions for second order elliptic equations with rapidly oscillating periodic coefficients in two-dimensional domain. We use an extension of the "mixed formulation" approach to obtain the representation formula satisfied by the oscillatory solution and homogenized solution by means of the particularity of solutions for equations in two-dimensional case. Then we utilize this formula in combination with the asymptotic estimates of Green or Neumann functions for operators and uniform regularity estimates of solutions to obtain convergence rates in L~p for solutions as well as gradient error estimates for Dirichlet or Neumann problems respectively.     

Continuous dependence results for non-linear Neumann type boundary value problems

We obtain estimates on the continuous dependence on the coefficient for second-order non-linear degenerate Neumann type boundary value problems. Our results extend previous work of Cockburn et al., Jakobsen and Karlsen, and Gripenberg to problems with more general boundary conditions and domains. A new feature here is that we account for the dependence on the boundary conditions. As one application of our continuous dependence results, we derive for the first time the rate of convergence for the vanishing viscosity method for such problems. We also derive new explicit continuous dependence on the coefficients results for problems involving Bellman-Isaacs equations and certain quasilinear equation.