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

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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.