arXiv:1908.03326v1 [math.NA] 9 Aug 2019 GALERKIN APPROXIMATION IN BANACH AND HILBERT SPACES W. ARENDT, I. CHALENDAR, AND R. EYMARD Abstract. In this paper we study the conforming Galerkin ap- proximation of the problem: find u ∈U such that a(u, v)= 〈L, v〉 for all v ∈V , where U and V are Hilbert or Banach spaces, a is a continuous bilinear or sesquilinear form and L ∈V ′ a given data. The approximate solution is sought in a finite dimensional subspace of U , and test functions are taken in a finite dimensional subspace of V . We provide a necessary and sufficient condition on the form a for convergence of the Galerkin approximation, which is also equiv- alent to convergence of the Galerkin approximation for the adjoint problem. We also investigate some connections between Galerkin approximation and the approximation property from geometry of Banach spaces. In the case of Hilbert spaces, we prove that the only bilinear or sesquilinear forms for which any Galerkin approx- imation converges (this property is called the universal Galerkin property ) are the essentially coercive forms. In this case, a gener- alization of the Aubin-Nitsche Theorem leads to optimal a priori estimates in terms of regularity properties of the right-hand side L, as shown by several applications. 1. Introduction Due to its practical importance, the approximation of elliptic prob- lems in Banach or Hilbert spaces has been the object of numerous works. In Hilbert spaces, a crucial result is the simultaneous use of the Lax-Milgram theorem and of C´ ea’s Lemma to conclude the con- vergence of conforming Galerkin methods in the case that the elliptic problem is resulting from a coercive bilinear or sesquilinear form. But the coercivity property is lost in many practical situations: for example, consider the Laplace operator perturbed by a convection term or a reaction term (see the example in Section 7.2), and the approxi- mation of non-coercive forms must be studied as well. For particular 2010 Mathematics Subject Classification. 65N30,47A07,47A52,46B20. Key words and phrases. Galerkin approximation, sesquilinear coercive forms, approximation properties in Banach spaces, essential coercivity, universal Galerkin convergence. 1