伪谱的边界曲线及其跟踪算法的步长控制

1 引言 1990年L.N.Trefethen为度量矩阵的非正规性引入了矩阵的伪谱,对于任意的ε≥0,矩阵A∈C~(n×n)的ε-伪谱定义为 ∧_ε(A)={z:z∈σ(A △A),||△A||≤ε},其中σ(·)表示矩阵的特征值集合(谱集合),||·||表示矩阵的2-范数。利用预解式(zI—A)~(-1),     

Some applications of a galerkin-collocation method for boundary integral equations of the first kind

Here we apply the boundary integral method to several plane interior and exterior boundary value problems from conformal mapping, elasticity and fluid dynamics. These are reduced to equivalent boundary integral equations on the boundary curve which are Fredholm integral equations of the first kind having kernels with logarithmic singularities and defining strongly elliptic pseudodifferential operators of order - 1 which provide certain coercivity properties. The boundary integral equations are approximated by Galerkin's method using B-splines on the boundary curve in connection with an appropriate numerical quadrature, which yields a modified collocation scheme. We present a complete asymptotic error analysis for the fully discretized numerical equations which is based on superapproximation results for Galerkin's method, on consistency estimates and stability properties in connection with the illposedness of the first kind equations in L₂. We also present computational results of several numerical experiments revealing accuracy, efficiency and an amazing asymptotical agreement of the numerical with the theoretical errors. The method is used for computations of conformal mappings, exterior Stokes flows and slow viscous flows past elliptic obstacles.     

A curve tracing algorithm for computing the pseudospectrum

The boundary curve of the pseudospectrum of a matrix is defined as a contour line of its resolvent norm. A rather simple and efficient continuation method is presented, which determines the implicitly given curve by a prediction-correction scheme, where the correction step is accomplished by one single Newton step. Besides its efficiency the algorithm turns out to be very accurate as long as the boundary of the pseudospectrum is a smooth curve. Problems may arise at bifurcation points where the resolvent norm is not differentiable.This work was partially supported by Deutsche Forschungsgemeinschaft.     

The coupling of bem and fem for the two-dimensional viscous flow problem

In this paper we propose a numerical scheme for treating the problem of sJow viscous flow past an obstacle in the plane. This scheme is a combination of boundary element and finite element methods. By introducing an auxiliary boundary curve, we divide the region under consideration into two subregions, an inner and an outer region. In the inner region, we employ a finite element method (FEM) for solving a system of simplified field equations with proper natural boundary conditions. In the outer region, the solution is expressed in the form of a simple-layer potential with density function satisfying a system of modified integral equations of the first kind. The latter are solved by a boundary element method (BEM). Both solutions are matched on the common auxiliary boundary curve. Error estimates in suitable function spaces are derived in terms of the mesh widths as well as the small parameters, the Reynolds numbers     

On chordal and bilateral SLE in multiply connected domains

We discuss the possible candidates for conformally invariant random non-self-crossing curves which begin and end on the boundary of a multiply connected planar domain, and which satisfy a Markovian-type property. We consider both, the case when the curve connects a boundary component to itself (chordal), and the case when the curve connects two different boundary components (bilateral). We establish appropriate extensions of Loewner’s equation to multiply connected domains for the two cases. We show that a curve in the domain induces a motion on the boundary and that this motion is enough to first recover the motion of the moduli of the domain and then, second, the curve in the interior. For random curves in the interior we show that the induced random motion on the boundary is not Markov if the domain is multiply connected, but that the random motion on the boundary together with the random motion of the moduli forms a Markov process. In the chordal case, we show that this Markov process satisfies Brownian scaling and discuss how this limits the possible conformally invariant random non-self-crossing curves. We show that the possible candidates are labeled by two functions, one homogeneous of degree zero, the other homogeneous of degree minus one, which describes the interaction of the random curve with the boundary. We show that the random curve has the locality property for appropriate choices of the interaction term. The research of the first author was supported by NSA grant H98230-04-1-0039. The research of the second author was supported by a grant from the Max-Planck-Gesellschaft.     

A discrete collocation method for Symm's integral equation on curves with corners

This paper is devoted to the approximate solution of the classical first-kind boundary integral equation with logarithmic kernel (Symm's equation) on a closed polygonal boundary in ℝ². We propose a fully discrete method with a trial space of trigonometric polynomials, combined with a trapezoidal rule approximation of the integrals. Before discretization the equation is transformed using a nonlinear (mesh grading) parametrization of the boundary curve which has the effect of smoothing out the singularities at the corners and yields fast convergence of the approximate solutions. The convergence results are illustrated with some numerical examples.     

Boundary Identification for a Blast Furnace

In this paper,the authors discuss an inverse boundary problem for the axi- symmetric steady-state heat equation,which arises in monitoring the boundary corrosion for the blast-furnace.Measure temperature at some locations are used to identify the shape of the corrosion boundary. The numerical inversion is complicated and consuming since the wear-line varies during the process and the boundary in the heat problem is not fixed.The authors suggest a method that the unknown boundary can be represented by a given curve plus a small perturbation,then the equation can be solved with fixed boundary,and a lot of computing time will be saved. A method is given to solve the inverse problem by minimizing the sum of the squared residual at the measuring locations,in which the direct problems are solved by axi- symmetric fundamental solution method. The numerical results are in good agreement with test model data as well as industrial data,even in severe corrosion case.     

The boundary element method with Lagrangian multipliers

On open surfaces, the energy space of hypersingular operators is a fractional order Sobolev space of order 1/2 with homogeneous Dirichlet boundary condition (along the boundary curve of the surface) in a weak sense. We introduce a boundary element Galerkin method where this boundary condition is incorporated via the use of a Lagrangian multiplier. We prove the quasi‐optimal convergence of this method (it is slightly inferior to the standard conforming method) and underline the theory by a numerical experiment. The approach presented in this article is not meant to be a competitive alternative to the conforming method but rather the basis for nonconforming techniques like the mortar method, to be developed. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

An artificial boundary method for the Hull-White model of American interest rate derivatives

The Hull-White (HW) model is a widely used one-factor interest rate model because of its analytical tractability on liquidly traded derivatives, super-calibration ability to the initial term structure and elegant tree-building procedure. As an explicit finite difference scheme, lattice method is subject to some stability criteria, which may deteriorate the computational efficiency for early exercisable derivatives. This paper proposes an artificial boundary method based on the partial differential equations (PDEs) to price interest rate derivatives with early exercise (American) feature under the HW model. We construct conversion factors to extract the market information from the zero-coupon curve and then reduce the infinite computational domain into a finite one by using an artificial boundary on which an exact boundary condition is derived. We then develop an implicit θ-scheme with unconditional stability to solve the PDE in the reduced bounded domain. With a finite computational domain, the optimal exercise strategy can be determined efficiently. Our numerical examples show that the proposed scheme is accurate, robust to the truncation size, and more efficient than the popular lattice method for accurate derivative prices. In addition, the singularity-separating technique is incorporated into the artificial boundary method to enhance accuracy and flexibility of the numerical scheme.     

两个各向同性半平面焊接的界面共线裂纹问题

复合材料焊接的界面裂纹问题是工程中碰到的实际问题,这方面已有不少有意义的研究工作（如献[3]-[6]等等）。本讨论两个各同性半平面焊接的界面共线裂纹问题,利用复变方法和解析函数边值问题题的基本理论,给出了弹性体应力分布封闭形式的解。     

一种改进的几何活动轮廓模型

针对传统的几何活动轮廓模型在分割具有凹形边界时,演化曲线不能准确定位的缺点,提出了一种改进的几何活动轮廓模型,该模型通过在原模型的演化方程中增加一个正的常数项,使得演化曲线在未到达目标边界时,加速曲线演化,而在到达边界附近时,该加速项逐渐减小为零,从而能够很好的完成对凹形边界的分割.实验证明,该方法不仅能够分割具有凹形边界的目标,还能够使演化过程加速,提高几何活动轮廓模型的分割速度.     

抛物化稳定性方程在可压缩边界层中应用的检验

用抛物化稳定性方程（PSE）,研究了可压缩边界层中扰动的演化,并与由直接数值模拟（DNS）所得进行比较．目的在检验PSE方法用于研究可压缩边界层中扰动演化的可靠性．结果显示,无论是亚音速还是超音速边界层,由PSE方法和由DNS方法所得结果都基本一致,而温度比速度吻合得更好．对超音速边界层,还计算了小扰动的中性曲线．与线性稳定性理论（LST）的结果相比,二者的关系和不可压边界层的情况相似．     

具有转向点曲线的椭圆型方程奇摄动边值问题

The singularly perturbed elliptic equation boundary value problem with a curve of turning point is considered. Using the method of multiple scales and the comparison theorem,the asymptotic behavior of solution for the boundary value problem is studied.     

Hadamard’s formula and couplings of SLEs with free field

The relation between level lines of Gaussian free fields (GFF) and SLE₄-type curves was discovered by O. Schramm and S. Sheffield. A weak interpretation of this relation is the existence of a coupling of the GFF and a random curve, in which the curve behaves like a level line of the field. In the present paper we study these couplings for the free field with different boundary conditions. We provide a unified way to determine the law of the curve (i.e. to compute the driving process of the Loewner chain) given boundary conditions of the field and to prove existence of the coupling. The proof is reduced to the verification of two simple properties of the mean and covariance of the field, which always relies on Hadamard’s formula and properties of harmonic functions. Examples include combinations of Dirichlet, Neumann and Riemann–Hilbert boundary conditions. In doubly connected domains, the standard annulus SLE₄ is coupled with a compactified GFF obeying Neumann boundary conditions on the inner boundary. We also consider variants of annulus SLE coupled with free fields having other natural boundary conditions. These include boundary conditions leading to curves connecting two points on different boundary components with prescribed winding as well as those recently proposed by C. Hagendorf, M. Bauer and D. Bernard.     

A structure-preserving finite element approximation of surface diffusion for curve networks and surface clusters

We consider the evolution of curve networks in two dimensions (2d) and surface clusters in three dimensions (3d). The motion of the interfaces is described by surface diffusion, with boundary conditions at the triple junction points lines, where three interfaces meet, and at the boundary points lines, where an interface meets a fixed planar boundary. We propose a parametric finite element method based on a suitable variational formulation. The constructed method is semi-implicit and can be shown to satisfy the volume conservation of each enclosed bubble and the unconditional energy-stability, thus preserving the two fundamental geometric structures of the flow. Besides, the method has very good properties with respect to the distribution of mesh points, thus no mesh smoothing or regularization technique is required. A generalization of the introduced scheme to the case of anisotropic surface energies and non-neutral external boundaries is also considered. Numerical results are presented for the evolution of two-dimensional curve networks and three-dimensional surface clusters in the cases of both isotropic and anisotropic surface energies.     

THE BOUNDARY VALUE PROBLEMS OF A CLASS OF ELLIPTIC EQUATION WITH A CURVE OF TURNING POINT

The boundary value problems of a class of elliptic equation with a curveof turning point are considered. Using the method of multiple scales and thecomparison theorem, the asymptotic behavior of solution for the boundary valueproblem is studied.     

具有转向点曲线的二阶椭圆型方程边值问题的奇摄动

本文讨论了;较一般的一类具有转向点曲线的奇摄动椭圆型方程的边值问题,利用多重尺度法和比较定理,研究了边值问题解的一致有效渐近近似式。     

Boundary value problems for first-order mixed complex equations with degenerate curve

This article deals with some boundary value problems for quasilinear mixed (elliptic-hyperbolic) complex equations of first order with degenerate hyperbolic curve in a simply connected domain. First, we give the representation theorem and prove the uniqueness of solutions for the above boundary value problems. Then using the method of successive iteration, the existence of solutions for the above problems is proved.     

小血管血流速度分布的一种计算方法

本文所提出的计算方法,其基础是对血液流动微连续统模型作了一种边界条件的改进,设想了血管内壁面上血细胞速度可能不为零.对于由Eringen所提出的关于刚性圆管中稳态血液流动方程,假设了血管内壁面上血细胞的旋转速度,及血细胞旋转速度分布曲线在管轴处的斜率,导出了计算血管中速度分布曲线的方法,并将按此理论计算而得的曲线与Bugliarello和Hayden在实验中测得的分布曲线及由Turk,Sylvester和Ariman所提出的计算公式的结果相比较.     

N-S方程的拟序扰动序列与改善的渐近展开匹配法

提出了以拟序扰动序列逼近N-S方程定解问题渐近解的一种方法。对N-S方程及其边界条件的渐近拟序扰动序列解进行了讨论,并应用此方法对球坐标系中的圆球绕流进行求解,改善了渐近展开匹配方法,使匹配函数更容易确定。改善后的阻力曲线与实测资料相比在雷诺数小于等于4×104以前完全吻合。