Research on mathematical analysis of finite difference methods and its numerical simulations based on fictitious domain and distributions

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1. Numerical calculation of finite difference methods based on fictitious domain and distributionsOur formulation for numerical calculation in finite difference method is to find the locally integrable function satisfying the partial differential equations in the sense of distribution. This idea leads naturally to the expression of Dirichlet-Neumann map on the fictitious boundary. We proved that numerical solutions defined by locally integrable functions are convergent to the genuine solution in the sense of the Schwartz distribution. Moreover we proposed new numerical algorithm which use the fast Fourier transform. This is based on the combination with the Fourier transform and the eigenvalues and eigenvectors of the coefficient matrix of Poisson's equation. From this idea we constructs a numerical method which we call the successive elimination of lines, and contribute to the numerical calculation for the transmission problem.2. Applied mathematical problem and its numerical simulationThe numerical method we developed here is possible to apply to various problems in environmental mathematics. One is the modeling and simulation for surface inversions which depend on the temperature and atmosphere in urban. And we showed the numerical simulation of the singular temperature in the town. Two is the research on the coastal pollution by spilled oil when it occurs in the case of the tanker accident. We showed it through the numerical simulation based on the fictitious domain method. Third is the numerical simulation in medical treatments. We showed a numerical simulation of the cerebrospinal fluid.As above mentiond, we researched mathematical analysis of finite difference methods from view points of fictitious domain and distribution, and established the numerical algorithm which is able to apply to various environmental mathematics. Our papers were published and research results were presented at international conferences.