Approximation in Banach Spaces by Galerkin Methods
In this chapter, we consider an abstract linear problem which serves as a generic model for engineering applications. Our first goal is to specify the conditions under which this problem is well-posed. We use the definition proposed by Hadamard [Had32]: a problem is well-posed if it admits a unique solution and if it is endowed with a stability property, namely the solution is controlled by the data. Two important results asserting well-posedness are presented: the Lax—Milgram Lemma and the Tanach—Nečas—Babuška Theorem. The former provides a sufficient condition for well-posedness, whereas the latter, relying on slightly more sophisticated assumptions, gives necessary and sufficient conditions. Then, we study approximation techniques based on the so-called Galerkin method. Both conformal and non-conformal settings are considered. We investigate under which conditions the stability properties of the abstract problem are transferred to the approximate problem, and we obtain a priori estimates for the approximation error. The last section of this chapter investigates a particular form of the Banach—Nečas—Babuška Theorem relevant to problems endowed with a saddle-point structure.
KeywordsBilinear Form Galerkin Method Reflexive Banach Space Approximation Setting Approximability Property
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