# Galerkin approximation of linear problems in Banach and Hilbert spaces

## TL;DR

This paper analyzes the convergence of Galerkin methods for linear problems in Banach and Hilbert spaces, establishing necessary and sufficient conditions and characterizing forms that guarantee convergence.

## Contribution

It provides a comprehensive characterization of Galerkin approximation convergence and identifies the forms that ensure universal convergence in Hilbert spaces.

## Key findings

- Necessary and sufficient condition for Galerkin convergence
- Characterization of forms with universal Galerkin property
- Optimal a priori estimates for coercive forms

## Abstract

In this paper we study the conforming Galerkin approximation of the problem: find u $\in$ U such that a(u, v) = <L, v> for all v $\in$ V, where U and V are Hilbert or Banach spaces, a is a continuous bilinear or sesquilinear form and L $\in$ 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 equivalent to convergence of the Galerkin approximation for the adjoint problem. We also characterize the fact that U has a finite dimensional Schauder decomposition in terms of properties related to the Galerkin approximation. In the case of Hilbert spaces, we prove that the only bilinear or sesquilinear forms for which any Galerkin approximation converges (this property is called the universal Galerkin property) are the essentially coercive forms. In this case, a generalization 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. Finally, a section entitled "Supplement" provides some consequences of our results for the approximation of saddle point problems.