Approximation in Banach Spaces by Galerkin Methods

Abstract

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 FormGalerkin MethodReflexive Banach SpaceApproximation SettingApproximability PropertyThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.