In accordance with the development of calculations, theoretical studies on the numerical scheme is now bearing fruit and much needed. In 1991 an article on the finite element method is applied to the problem of evolution was published. After the method, basically this book study the various schemes from the point of view of operator theory. Many sections are devoted to the finite element method, but other schemes and the problem (cost simulation methods, domain decomposition method, nonlinear problems, and so on) are also discussed, motivated by the observation that practically useful scheme has a mathematical structure smooth and converses are also true. Boundary Value Problems and FEM. Other methods in the Discretization of Time. Other methods in the Discretization of Space. Domain Decomposition Method.