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Recent developments in numerical computations are amazing. A lot of huge projects in applied and theoretical sciences are becoming successful by them, while similar things are happening even in the level of personal computers. Under such a situation, theoretical studies on numerical schemes are fruitful and highly needed. The purpose of the present book is to provide some of them, particularly for schemes to solve partial differential equations. In 1991, we published an article on the finite element method applied to evolutionary problems from Elsevier Publishers (Fujita and Suzuki [148]). This book follows basically that way of description. We study various schemes from the operator theoretical point of view. Many parts are devoted to the finite element method, of which history is described in Oden [306]. We deal with elliptic and then time dependent problems in use of the semigroup theory and so forth. Some other schemes and problems are also discussed, with the later development taken also into account. We are led to believe that any scheme used practically has significant and tight structures mathematically and the converse is also true.

Operator theory

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Operator theory

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Exact eigenvalues, eigenvectors, and principal vectors of operators with infinite dimensional ranges can rarely be found. Therefore, one must approximate such operators by finite rank operators, then solve the original eigenvalue problem approximately. This book addresses this issue of solving eigenvalue problems for operators on infinite dimensional spaces. From a review of classical spectral theory, through approximation techniques, to ideas for further research that would extend the results described, this volume serves as both a text for graduate students and as a source of state-of-the-art results for research scientists.

Operator theory. --- Spectral theory (Mathematics).

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In accordance with the developments in computation, theoretical studies on numerical schemes are now fruitful and highly needed. In 1991 an article on the finite element method applied to evolutionary problems was published. Following the method, basically this book studies various schemes from operator theoretical points of view. Many parts are devoted to the finite element method, but other schemes and problems (charge simulation method, domain decomposition method, nonlinear problems, and so forth) are also discussed, motivated by the observation that practically useful schemes have fine

Numerical analysis. --- Operator theory. --- Functional analysis --- Mathematical analysis

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Operator theory --- Dirichlet problem --- Parabolic operators --- Operators, Parabolic --- Partial differential operators --- Boundary value problems --- Dirichlet problem. --- Dirichlet, Problème de.