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 mathematical structures and the converses are also true.
The present chapter is concerned with the finite element method (FEM) applied to elliptic boundary value problems. For this topic, we have several monographs such as Strang and Fix [359], Ciarlet [83], Raviart and Thomas [326], Johnson [193], Ciarlet and Lions [85], SzabĂł and BabuĆĄuka [372], and Brenner and Scott [60]. Here, we describe it in the framework of operator theory, picking up approximate operators of the schemes. This way is natural and efficient, particularly in dealing with time dependent problems, because the finite element method is regarded as a discretization of the underlying variational structure.
