Titlebook: Numerical Analysis; Roger Temam Book 1973 D. Reidel Publishing Company, Dordrecht, Holland 1973 Approximation.calculus.finite element meth

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Spaces of Functions Associated with an open set in RLet Ω be an open set of R. with boundary Γ; we will associate with this open set several spaces of functions for later use.

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Approximation of Some Function Spaces by Finite Differences (II)In this chapter we construct an internal approximation of .(Ω), an external approximation of .(Ω) and a generalized external approximation of .(Ω), using finite differences. We keep the notations introduced in Section 8.1, and the open set Ω is assumed to be bounded.

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Example II: The Neumann ProblemWe are in the situation discussed in Section 1.2; Ω is a bounded open set in R. with boundary Г, we put .(Ω) and .(Ω) and these spaces are provided with their usual Hubert structure (cf. Chapter 7):

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The Exact ProblemWe will describe here the nonlinear elliptic problem that we want to study, and we prove existence and uniqueness of the solutions of this problem. Existence is shown by the Galerkin method, which at the same time gives first method for approximate solution of the equation.

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Approximate ProblemsWe study the approximate solution of problem (13.1) by a finite difference method.

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