高波数波动问题的多水平方法

本文主要讨论求解高波数Helmholtz方程的多水平方法,主要回顾了一些具有代表性的多重网格方法.如Erlangga等人的shifted Laplacian预处理的多重网格法;Elman等提出的修正的多重网格方法;以及我们的基于连续内罚有限元(CIP-FEM)离散代数系统的多水平算法.最后还介绍了求解高波数时谐Maxwell方程的CIP-FEM离散代数系统的多水平算法.     

二维Helmholtz方程不适定问题的一种算子软化正则法

本文研究带有混合边界的二维Helmholtz方程不适定问题.为了获得稳定的数值解，利用基于de la ValléePoussin算子的软化正则方法，得到了正则近似解，给出正则近似解与精确解之间在先验参数选取规则之下的误差估计，并通过数值实验检验了数据有噪声扰动时方法的有效性和稳定性.     

Error Estimation and Optimization of the Functional Algorithms of a Random Walk on a Grid Which Are Applied to Solving the Dirichlet Problem for the Helmholtz Equation

We consider the algorithms of a random walk on a grid which are applied to global solution of the Dirichlet problem for the Helmholtz equation (the direct and conjugate methods). In the metric space C we construct some upper error bounds and obtain optimal values (in the sense of the error bound) of the parameters of the algorithms (the number of nodes and the sample size).     

Normalized system of functions with respect to the Laplace operator and its applications

A system of functions 0-normalized with respect to the operator Δ in some domain is constructed. Application of this system to boundary value problems for the polyharmonic equation is considered. Connection between harmonic functions and solutions of the Helmholtz equation is investigated.     

Ion redistribution in an electric double layer

The structure of a single flat electric double layer (EDL) is studied by grounding a symmetric electrolyte (NaCl), which is in contact with a planar positively corona-treated polypropylene film. Because the profiles of the electrostatic potential distribution and ion distribution in the solution are altered when the solution is grounded, some mobile counterions in the diffuse layer of the electrolyte solution will go into the Helmholtz layer and thus decrease the electric potential psi(a/2) at the Stern plane in order to obtain a new equilibrium. After the system is grounded for a long time, the representation of the electric double layer changes from a Stern model to a Helmholtz model. Theoretical and experimental analyses are given in this study.     

Mathematical Modeling of Near-Field Optics

Consider a time-harmonic electromagnetic plane wave incident on a periodic (grating) structure. An inhomogeneous (subwavelength) object is placed inside the periodic structure. The scattering problem is to study the electromagnetic field distributions. The problem arises in the study of near-field optics and has many physical and biological applications. This work is devoted to modeling and analysis of a near-field optics problem. An integral representation approach is presented to solve the problem. It is shown particularly that the perturbation due to the object decays exponentially along the periodic direction of the structure, provided that no surface waves occur. Based on the approach, a general solution method is discussed and the well-posedness of the model problems are established.     

Modelling aqueous solutions near the critical point of water

We review the thermodynamic properties of dilute solution near the critical point of the solvent. Two examples are discussed, a solution of a non-electrolyte and a solution of an electrolyte. The limiting behavior of the electrolyte solutions is modelled with a Debye-Huckel term in the Helmholtz free energy. The partial molar properties, in particular the volume and isobaric thermal expansion are examined in detail. The derivation of these properties is introduced by considering the geometry of the thermodynamic surfaces near to and far from the critical point of the solvent. We conclude that the properties of solutions near the solvent critical point are dominated by that feature; solution properties cannot be adequately modelled without including the functional forms associated with the critical point.     

A Spectral Element/Laguerre Coupled Method to the Elliptic Helmholtz Problem on the Half Line

A Legendre spectral element/Laguerre coupled method is proposed to numerically solve the elliptic Helmholtz problem on the half line. Rigorous analysis is carried out to establish the convergence of the method. Several numerical examples are provided to confirm the theoretical results. The advantage of this method is demonstrated by a numerical comparison with the pure Laguerre method.     

Kantorovich solution for the problem of bending of a ladder plate

Based on the Kantorovich approximation solution for a rectangular plate in bending, this paper deals with the solutions for the ladder plate with various boundary conditions. The deflection of the plate is expressed in a first-order displacement function w(x,y)=u(x,y)v(y) where the u(x,y) in x direction is the generalized beam function. By making use of the principle of least potential energy, the variable coefficients differential equations for v(y) may be established. By solving is, these differential eugations and making use of the boundary conditions, the accurate solutions of v(y) in y direction may be obtained. Then the displacement function w(x,y) is the solution for the problem of the bending of the ladder plate with a better degree of approximation.     

Helmholtz方程在多层无界声波导中特征问题计算

尝试在深度方向引入PML（完美的匹配层）选取适当的非线性局部坐标．tY-交变换及方程变换,将Helmholtz方程uxx＋uzz＋k^2（x,z）u=0变换为Vxx＋aVzz＋βVz＋rV=0,然后再选取非对称Rayleigh商迭代算法来求解Helmholtz方程的特征问题,即求解aδz^2＋βδz＋r的特征值和特征函数．数值模拟表明,此方法是可行的,并可以大大改善数值求解的效率．