Updating of Poisson Solvers

1. 在用有限元法计算翼型周围二维完美流体时，设置一个足够大的包含翼型的圆盘，取圆盘与翼型的补部相交的双连通区域作为我们的计算域。在计算域的圆形边界上设置透明边界条件。在计算域的圆形边界上设置透明边界条件，对域内离散拉普拉斯问题进行有限元数值计算。为了解决离散化问题，提出了一种二维外部拉普拉斯问题的FEM-CSM（有限元法-电荷模拟法）组合方法，并进行了数学验证。该方法采用电荷模拟的方法逼近透明边界条件。对于圆盘外域Helmholtz方程的一种FSM（基本解法）近似问题，给出了源点与配点等距等相位排列情况下的唯一可解定理和误差估计定理。研究者的工作取得了显著进展：(1)奇、偶维Radon变换的反演表示和基于它们的数值重建程序（J. Watanabe）；(2)非对称Abel反演中的神经网络计算（T. Takeda和M . Fukuhara）；(3)基于域分解法和虚域法的有限元数值方法应用于无界区域辐射和散射问题（T. Kako）。(4)基于时滞微分方程的动力系统数值模拟的基础研究（T. Koto）。(5)一类椭圆型偏微分算子的特征值问题的自由边界问题（I. Ohnishi）。(6)描述螺旋位错在晶体表面生长的反应扩散方程及其奇异极限的理论研究（由K. Nakamura）。(7)三维外Helmholtz问题的有限元计算（D. Koyama）。(8)地幔对流三维数值模拟并行计算代码（M. Tabata）。(9)利用电荷模拟方法提出了无界多连通域到正则狭缝域的数值共形映射方法，并通过数值实验验证了该方法的有效性（K. Amano）。（10）基于sinc函数的一维泊松解的理论研究（M. Sugihara）。（11）残余切削法的数学基础（高桥T.）。（12）泊松求解器在生物信息学中的作用（作者：S. Ihara）。（13）纳米技术结构形成中多相流的基本方程组的推导（作者：H. Yasuda）。少

1. Main Results Obtained in the Joint Works around the Head Investigator :In the computatioin of two dimensional perfect fluid around an air foil through finite element method, setting a sufficiently large disc containing the air foil, let a doubly connected region, which is the intersection of the disc and the complement of the air foil, be our computational domain. Setting a transparent boundary condition on the circular boundary of the computational domain. Setting a transparent boundary condition on the circular boundary of the computational domain, we execute finite element numerical computations for discretized Laplace problems in the domain. To solve the discretized problems, an FEM-CSM (finite Element Method -- Charge Simulation Method) combined method for 2D exterior Laplace problems is proposed and mathematically justified. In this combined method transparent boundary condition is approximated through charge simulation method. For an FSM (Fundamental Solution Method) approxim … More ate problem for Helmholtz equation in the exterior domain of a disc, a unique solvability theorem and an error estimate theorem are obtained in the case of equi-distant equally phased arrangement of source points and collocation points.2. Remarkable Progress Obtained in the Works by Investigators :(1) Representations of the inversions of the odd and even dimensional Radon transforms and numerical reconstruction procedures based on them (by J. Watanabe)(2) Neural network calculation in asymmetric Abel inversion (by T. Takeda and M Fukuhara)(3) Finite element numerical method based on the domain decomposition method and fictitious domain method applied to radiation and scattering problem in unbounded region (by T. Kako).(4) fundamental research for numerical simulation of dynamical systems generated by delay differential equations (by T. Koto).(5) A free boundary problem related to eigenvalue problems for a class of elliptic partial differential operators (by I. Ohnishi).(6) Theoretical study on reaction-diffusion equations and their singular limits describing the growth of screw dislocation on the crystal surface (by K. Nakamura).(7) Finite element computations for 3D exterior Helmholtz problem (by D. Koyama).(8) A parallel computation code for 3D numerical simulation of Earth's mantle convection (by M. Tabata).(9) Proposal of a method of numerical conformal mappings of unbounded multiply-connected domains onto canonical slit domains using the charge simulation method, and confirmation of its effectiveness by numerical experiments (by K. Amano).(10) A theoretical study on the 1-dimensional Poisson solvers based on sinc functions (by M. Sugihara).(11) Mathematical foundation of residual cutting method (by T. Takahashi).(12)Role of Poisson solvers in bio-informatics (by S. Ihara).(13) Derivation of basic system of equations for multi phase flow appearing in the structure formation in nanotechnology (by H. Yasuda). Less