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.

我们将边界元素方法（BEM）的收敛性和稳定性的数学研究作为椭圆边界值问题的求解器。我们还为我们的问题提供了数值研究，并在本研究中开发了多重复系统的计算环境。根据边界元素方法和边界积分方程方法的传统研究，我们重点估计了融合研究的界限，但我们指出了缺乏传统研究的重要性，并集中了对数字解决方案的估计的重要性。我们通过BEM表现出比域相对于域的高精度，并阐明了BEM的优点之一。我们可以在域中的紧凑型集合上观察到数值溶液的准确均匀收敛，并且域中的收敛速率高于边界的平滑数据。我们还观察到，即使数值解决方案不在边界上均匀收敛，数值溶液的均匀收敛也是如此。该功绩在数值研究中利用了与椭圆形偏微分方程相关的不足问题的优势。每个研究者在主管研究员的控制下都分别进行了研究。首席研究员及其研究小组研究BEM的数值实验应用于典型的椭圆边界值问题，以显示BEM的高精度现象。在本研究中，他们还开发了非常快速的多重复系统。其他研究人员主要处理BEM和其他研究人员主要研究反问题。开发的多重复系统可以有力地适用于数值分析的广泛领域。

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)。

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)。

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。

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)。