Mathematical Study of the Boundary Element Method and its Application to Inverse

边界元法的数学研究及其在反演中的应用

• 批准号: 10490018
• 负责人: ISO Yuusuke
• 金额: $ 7.74万
• 依托单位: KYOTO UNIVERSITY
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (B)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 1999
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10490018/
• 关键词: Boundary Element Method; BEM; Numerical Analysis; Applied Analysis; Ill-posed Problems; Inverse Problems; Multiprecision Computations; Boundary Integral Equation

We deal with mathematical study of convergence and stability for the boundary element method (BEM) as a solver for elliptic boundary value problems. We also give numerical study for our problems, and we develop the computational environment of multiprecision system in the present research.According to the traditional study for the boundary element method and the boundary integral equation method, we have focused estimation on boundaries in the study of the convergence, but we pointed out the lack of the traditional studies and focused importance of estimation for numerical solution over domains in the first step. We show high accuracy of numerical solutions over domain by BEM, and we clarify one of the merits of BEM in the research. We can observe accurate uniform convergence of numerical solution on a compact set in the domain, and convergence rate in the domain is higher than that on the boundary for smooth data. We also observe, and uniform convergence of numerical solution on a compact set even when numerical solution do not converge uniformly on the boundary. The merit takes advantage in the numerical study for ill-posed problems connected with elliptic partial differential equations.The research has been carried out separately by each investigator under the control of the head investigator. The head investigator and his research group study numerical experiments of BEM applied to typical elliptic boundary value problems to show high accuracy phenomena of BEM. And they also develop very fast multiprecision system in the present research. The other investigators deal mainly with BEM and others mainly study inverse problems. The multiprecision system developed can be applicable powerfully in the vast fields of numerical analysis.

本文研究了边界元法求解椭圆边值问题的收敛性和稳定性。本文还对问题进行了数值研究，发展了多精度系统的计算环境，在传统的边界元法和边界积分方程法研究的基础上，我们在收敛性研究中着重于边界估计，但首先指出了传统研究的不足，并着重指出了区域上数值解估计的重要性。通过对边界元法在区域上数值解的高精度分析，阐明了边界元法在研究中的优点之一。我们可以观察到数值解在区域内的紧集上的精确一致收敛，并且对于光滑数据，区域内的收敛速度比边界上的收敛速度快。我们还观察到，即使数值解在边界上不一致收敛，数值解在紧集上也一致收敛。其优点是在椭圆型偏微分方程不适定问题的数值研究中发挥了优势，研究工作在首席研究员的领导下由每个研究员单独进行。本课题组主要研究边界元法应用于典型椭圆边值问题的数值实验，以展示边界元法的高精度现象。在目前的研究中，他们还开发了非常快速的多精度系统。其他研究者主要研究边界元法，其他研究者主要研究反问题。所开发的多精度系统可在数值分析的广阔领域中得到有力的应用。

项目成果

大西 和榮: "「An approximate variational method for the Cauchy problem in plane elastostatics」" Theoretical and Applied Mechanics. 47. 341-347 (1998)

Kazuei Onishi：“平面弹性静力学中柯西问题的近似变分方法”理论与应用力学 47. 341-347 (1998)。

Y.Ohura, K.Kobayashi, K, Onishi: "Numerical solution of an under-deteremined problem of the Laplace equation"Journal of Applied Mechanics, JSCE. 2. 185-189 (1999)

Y.Ohura、K.Kobayashi、K、Onishi：“拉普拉斯方程欠定问题的数值解”应用力学杂志，JSCE。

Krishna M.Singh, Masataka Tanaka: "Dual reciprocity boundary element analysis of nonlinear diffusion : temporal discretization"Engineering Analysis with Boundary Elements. 23. 419-433 (1999)

Krishna M.Singh、Masataka Tanaka：“非线性扩散的双互易边界元分析：时间离散化”边界元工程分析。

久保 司郎: "「A Mathematical and Numerical Study on Regularization of an Inverse Boundary Value」" Inverse Problems in Engineering Mechanics. 337-344 (1998)

Shiro Kubo：“‘逆边界值正则化的数学和数值研究’”工程力学逆问题337-344 (1998)。

岩野 功,磯 祐介 他: "Laplace 方程式の BEM 解析における収束評価の精密化について"境界要素法論文集. 16巻. 31-36 (1999)

Isao Iwano、Yusuke Iso 等人：“改进拉普拉斯方程 BEM 分析中的收敛性评估”边界元方法论文集 16. 31-36 (1999)。