Generation of structured difference grids in two-dimensional nonconvex domains using mappings

Generation of structured difference grids in two-dimensional nonconvex domains is considered using a mapping of a parametric domain with a given nondegenerate grid onto a physical domain. For that purpose, a harmonic mapping is first used, which is a diffeomorphism under certain conditions due to Rado’s theorem. Although the harmonic mapping is a diffeomorphism, its discrete implementation can produce degenerate grids in nonconvex domains with highly curved boundaries. It is shown that the degeneration occurs due to approximation errors. To control the coordinate lines of the grid, an additional mapping is used and universal elliptic differential equations are solved. This makes it possible to generate a nondegenerate grid with cells of a prescribed shape.     

有界多连通区域数值保角变换的GMRES(m)法北大核心CSCD

求解复杂多连通区域的保角变换函数是困难的.针对这一问题,该文将求解保角变换函数转化为利用模拟电荷法求解一对定义在问题区域上的共轭调和函数,再根据边界条件建立约束方程,并利用GMRES(m)(the generalized minimal residual method)算法求解约束方程,获得了模拟电荷,进而构造了高精度的近似保角变换函数,将有界多连通区域映射为三种无界正则狭缝域.数值实验验证了该文算法的有效性.     

Harmonic mappings onto stars

A general version of the Radó-Kneser-Choquet theorem implies that a piecewise constant sense-preserving mapping of the unit circle onto the vertices of a convex polygon extends to a univalent harmonic mapping of the unit disk onto the polygonal domain. This paper discusses similarly generated harmonic mappings of the disk onto nonconvex polygonal regions in the shape of regular stars. Calculation of the Blaschke product dilatation allows a determination of the exact range of parameters that produce univalent mappings.     

On a Problem of the Constructive Theory of Harmonic Mappings

The problem of irremovable error appears in finite difference realization of the Winslow approach in the constructive theory of harmonic mappings. As an example, we consider the well-known Roache–Steinberg problem and demonstrate a new approach, which allows us to construct harmonic mappings of complicated domains effectively and with high precision. This possibility is given by the analytic-numerical method of multipoles with exponential convergence rate. It guarantees effective construction of a harmonic mapping with precision controlled by an a posteriori estimate in a uniform norm with respect to the domain.     

Local defect correction with slanting grids

The local defect correction (LDC) method is used to solve a convection‐diffusion‐reaction problem that contains a high‐activity region in a relatively small part of the domain. The improvement of the solution on a coarse grid is obtained by introducing a correction term computed from a local fine‐grid solution. This article studies problems where the high‐activity region is covered with a rectangular fine grid not aligned with the axes of the global domain. This study shows that the resulting method is less expensive than both a uniform refinement and tensor product grid method. © 2003 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 20: 1–17, 2004.     

Two‐grid methods for time‐harmonic Maxwell equations

In this paper, we develop several two‐grid methods for the Nédélec edge finite element approximation of the time‐harmonic Maxwell equations. We first present a two‐grid method that uses a coarse space to solve the original problem and then use a fine space to solve a corresponding symmetric positive definite problem. Then, we present two types of iterative two‐grid methods, one is to add the kernel of the curl ‐operator in the fine space to a coarse mesh space to solve the original problem and the other is to use an inner iterative method for dealing with the kernel of the curl ‐operator in the fine space and the coarse space, separately. We provide the error estimates for the first two methods and present numerical experiments to show the efficiency of our methods.Copyright © 2012 John Wiley & Sons, Ltd.     

Quasiconformal harmonic mappings between \fancyscriptC1,m{{\fancyscript{C}}^{1,\mu}} Euclidean surfaces

The conformal deformations are contained in two classes of mappings quasiconformal and harmonic mappings. In this paper we consider the intersection of these classes. We show that, every K quasiconformal harmonic mapping between surfaces with boundary is a Lipschitz mapping. This extends some recent results of several authors where the same problem has been considered for plane domains. As an application it is given an explicit Lipschitz constant of normalized isothermal coordinates of a disk-type minimal surface in terms of boundary curve only. It seems that this kind of estimates are new for conformal mappings of the unit disk onto a Jordan domain as well.     

Debye sources and the numerical solution of the time harmonic Maxwell equations

In this paper, we develop a new representation for outgoing solutions to the time‐harmonic Maxwell equations in unbounded domains in ?³. This representation leads to a Fredholm integral equation of the second kind for solving the problem of scattering from a perfect conductor, which does not suffer from spurious resonances or low‐frequency breakdown, although it requires the inversion of the scalar surface Laplacian on the domain boundary. In the course of our analysis, we give a new proof of the existence of nontrivial families of time‐harmonic solutions with vanishing normal components that arise when the boundary of the domain is not simply connected. We refer to these as k‐Neumann fields, since they generalize, to nonzero wave numbers, the classical harmonic Neumann fields. The existence of k‐Neumann fields was established earlier by Kress. © 2009 Wiley Periodicals, Inc.     

调和单叶映射反函数调和的充要条件

给出复值，调和，单叶函数的反函数是调和单叶的充要条件；并且举例说明反函数一     

Local defect correction with different grid types

For a Poisson problem with a solution having large gradients in (nearly) circular subregions a local defect correction method is considered. The problem on the global domain is discretized on a cartesian grid, whereas the restriction of the problem to a circular subdomain is discretized on a polar grid. The two discretizations are then combined in an iterative way. We show that LDC can be viewed as an iterative method for the Poisson equation on a single composite cartesian‐polar grid. The efficiency of methods is illustrated by numerical examples. © 2002 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 18: 454–468, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/num.10018     

The domain derivative of time‐harmonic electromagnetic waves at interfaces

The scattering of time‐harmonic electromagnetic waves by a penetrable obstacle is considered. In view of shape optimization or inverse reconstruction problems, the domain derivative of the scattering problem is investigated. Existence of the derivative in the sense of a Fréchet derivative and a characterization by a transmission boundary value problem are shown. Copyright © 2012 John Wiley & Sons, Ltd.     

Multi‐grid for PDEs on complicated domains

A multi‐grid method for solving linear equation systems as they arise from finite element discretisations of elliptic boundary value problems on complicated domains will be presented and analysed. The emphasis is on the robustness of this algorithm with respect to the geometric details in the domain. Robust multi‐grid convergence can be proved for a two‐dimensional model problem.     

Convergence of the solution for a domain decomposed,heterogeneously modeled flow past a concave profile problem

The free streamline problem investigated is that of fluid flow past a symmetric truncated concave‐shaped profile between walls. An open wake or cavity is formed behind the profile. Conformal mapping techniques are used to solve this problem. The problem formulated in the hodograph plane is decomposed into two nonoverlapping domains. Heterogeneous modeling is then used to describe the problems, i.e., a different governing differential equation in each domain. In one of these domains, a Baiocchi‐type transformation is used to obtain a fixed domain formulation for the part of the transformed problem containing an unknown boundary. In the other domain, the Baiocchi‐type transformation is extended across the boundary between the two domains, thus yielding a different problem formulation. This also assures that the dependent variables and their normal derivatives are continuous along this common boundary. The numerical solution scheme, a successive over‐relaxation approach, is applied over the whole problem domain with the use of a projection‐operation over only the fixed domain formulated part. Numerical results are obtained for the case of a truncated circular profile. These results are found to be in good agreement with another published result. The existence and uniqueness of the solution to the problem as a variational inequality is shown, and the convergence of the numerical solution using a domain decomposition method scheme is demonstrated by assuming some convergence property on the common interface of the two subdomains. © 2000 John Wiley & Sons, Inc. Numeer Methods Partial Differential Eq 16: 459–479, 2000     

Doubly Connected Minimal Surfaces and Extremal Harmonic Mappings

The concept of a conformal deformation has two natural extensions: quasiconformal and harmonic mappings. Both classes do not preserve the conformal type of the domain, however they cannot change it in an arbitrary way. Doubly connected domains are where one first observes nontrivial conformal invariants. Herbert Gr?tzsch and Johannes C.C.?Nitsche addressed this issue for quasiconformal and harmonic mappings, respectively. Combining these concepts we obtain sharp estimates for quasiconformal harmonic mappings between doubly connected domains. We then apply our results to the Cauchy problem for minimal surfaces, also known as the Bj?rling problem. Specifically, we obtain a sharp estimate of the modulus of a doubly connected minimal surface that evolves from its inner boundary with a given initial slope.     

加权调和Bergman核及其诱导的度量矩阵

在Rn中的有界域上建立加权调和Bergman核,并得出单位球的加权调和Bergman核的表达式;利用加权调和Bergman核在Rn的有界域上构造度量矩阵;得到关于调和映射的Jacobi矩阵与度量矩阵之间的一个不等式.     

The Schwarzian Derivative of Harmonic Mappings in the Plane

In this paper, the authors introduce a definition of the Schwarzian derivative of any locally univalent harmonic mapping defined on a simply connected domain in the complex plane. Using the new definition, the authors prove that any harmonic mapping f which maps the unit disk onto a convex domain has Schwarzian norm ■≤ 6. Furthermore, any locally univalent harmonic mapping f which maps the unit disk onto an arbitrary regular n-gon has Schwarzian norm■.     

On planar harmonic grid generation

It has been shown that the harmonic map is a diffeomorphism. In some cases, numerical solutions to the equation have been noticed to produce folded grids, however. The folding of the grid is due to truncation error and not an incorrect theorem as has been suggested earlier. Difference approximations are constructed that solve the problem. The difference approximations can be of low order, which shows that the order of the difference approximations is not the key factor. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 305–315, 1999     

Boundary‐conforming coordinates with grid line control for acoustic scattering from complexly shaped obstacles

The current work sets forth a practical approach to numerically solve two‐dimensional direct acoustic scattering problems from complexly shaped scatterers with severe singularities, such as corners and cusps. First, boundary conforming coordinates are generated. This generation is performed through an elliptic grid generator algorithm, including control of the coordinate lines. The grid line control solely depends on the initial distribution of grid points. Following the grid generation process, the initial boundary value problem, modelling the scattering phenomenon, is formulated in terms of the new curvilinear coordinates, and a finite‐difference time domain method is implemented. The presence of the boundary singularities causes instability of the numerical method. However, by appropriately controlling the distance between grid lines in the vicinity of these singularities, stability and convergence are achieved. A semianalytical formula for the differential scattering cross‐section is obtained from the discrete Fourier transform of the computed scattered pressure field. The method is successfully applied to several interesting scatterers of various shapes. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

The point source method for inverse scattering in the time domain

Many recent inverse scattering techniques have been designed for single frequency scattered fields in the frequency domain. In practice, however, the data is collected in the time domain. Frequency domain inverse scattering algorithms obviously apply to time‐harmonic scattering, or nearly time‐harmonic scattering, through application of the Fourier transform. Fourier transform techniques can also be applied to non‐time‐harmonic scattering from pulses. Our goal here is twofold: first, to establish conditions on the time‐dependent waves that provide a correspondence between time domain and frequency domain inverse scattering via Fourier transforms without recourse to the conventional limiting amplitude principle; secondly, we apply the analysis in the first part of this work toward the extension of a particular scattering technique, namely the point source method, to scattering from the requisite pulses. Numerical examples illustrate the method and suggest that reconstructions from admissible pulses deliver superior reconstructions compared to straight averaging of multi‐frequency data. Copyright © 2006 John Wiley & Sons, Ltd.