Galerkin approximation of linear problems in Banach and Hilbert spaces

Author:

Arendt W1,Chalendar I2,Eymard R2

Affiliation:

1. Institute of Applied Analysis, University of Ulm. Helmholtzstr. 18, D-89069 Ulm, Germany

2. LAMA, Univ. Gustave Eiffel, Univ. Paris Est Créteil, CNRS, F-77454 Marne-la-Vallée, France

Abstract

Abstract In this paper we study the conforming Galerkin approximation of the problem: find $u\in{{\mathcal{U}}}$ such that $a(u,v) = \langle L, v \rangle $ for all $v\in{{\mathcal{V}}}$, where ${{\mathcal{U}}}$ and ${{\mathcal{V}}}$ are Hilbert or Banach spaces, $a$ is a continuous bilinear or sesquilinear form and $L\in{{\mathcal{V}}}^{\prime}$ a given data. The approximate solution is sought in a finite-dimensional subspace of ${{\mathcal{U}}}$, and test functions are taken in a finite-dimensional subspace of ${{\mathcal{V}}}$. 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 ${{\mathcal{U}}}$ 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.

Funder

Agence Nationale de la Recherche

Publisher

Oxford University Press (OUP)

Subject

Applied Mathematics,Computational Mathematics,General Mathematics

Reference32 articles.

1. The Dirichlet-to-Neumann operator via hidden compactness;Arendt;J. Funct. Anal.,2014

2. Partielle Differenzialgleichungen

3. Error-bounds for finite element method;Babuška;Numer. Math.,1970/71

4. On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers;Brezzi;Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge,1974

5. Springer Series in Computational Mathematics;Brezzi,1991

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