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Operator Theory and Numerical Methods
H. Fujita, N. Saito, T. Suzuki
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eBook - ePub
Operator Theory and Numerical Methods
H. Fujita, N. Saito, T. Suzuki
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About This Book
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.
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Topic
MathematicsSubtopic
Mathematical AnalysisChapter 1
Elliptic Boundary Value Problems and FEM
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.
1.1 Elliptic Boundary Value Problems
To fix the idea, let Ω â 2 be a convex polygon, and consider the Poisson equation
with Dirichlet condition
This boundary value problem has the weak form, described under the following notations.
1°. LP(Ω) denotes the set of P-integrable functions for P â [1, â) and that of essentially bounded functions for P = â. Unless otherwise stated we assume functions to be real-valued and Banach spaces real.
2°. Wm,p(Ω) denotes the Sobolev space, the ...