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

讨论了边界元方法(BEM)作为椭圆边值问题求解器的收敛和稳定性的数学研究。根据传统的边界元方法和边界积分方程法的研究，我们在收敛的研究中侧重于边界估计，但我们指出了传统研究的不足，并在第一步强调了区域数值解估计的重要性。我们证明了边界元在区域上数值解的高精度，并阐明了边界元在研究中的优点之一。我们可以观察到数值解在区域紧集上的精确一致收敛，并且对于光滑的数据，区域内的收敛速度高于边界上的收敛速度。当数值解在边界上不一致收敛时，我们也观察到数值解在紧集上的一致收敛。这一优点在与椭圆型偏微分方程相关的不适定问题的数值研究中具有优势，每个调查者在首席调查者的控制下分别进行研究。首席研究员和他的研究小组研究了边界元应用于典型椭圆型边值问题的数值实验，以显示边界元的高精度现象。在目前的研究中，他们还开发了非常快的多精度系统。其他研究者主要研究边界元问题，其他研究者主要研究逆问题。所开发的多精度系统在数值分析的广阔领域具有很强的适用性。

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.