Galerkin Approximation In Banach and Hilbert Spaces
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In this paper we study the conforming Galerkin approximation 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 equivalent 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 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.
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