Banach和希尔伯特空间中线性问题的Galerkin逼近
Galerkin approximation of linear problems in Banach and Hilbert spaces
摘要：
本文研究问题的协调Galerkin逼近：求u $\in$ U使得对所有v $\in$ V，a（u，v）= <L，v>，其中U和V是Hilbert或巴拿赫空间，a是连续双线性或半双线性型，L $\in$ V'是给定数据.在U的有限维子空间中求近似解，在V的有限维子空间中取检验函数，给出了Galerkin逼近收敛的形式a的充要条件，它等价于伴随问题的Galerkin逼近收敛.我们还刻画了这样一个事实，即U有一个有限维Schauder分解的性质有关的Galerkin逼近。在希尔伯特空间的情形下，我们证明了任何Galerkin逼近对其收敛的唯一双线性或半双线性型（这个性质称为泛Galerkin性质）是本质强制型.在这种情况下，Aubin-Nitsche定理的推广导致最佳的先验估计的右手边L的正则性，如几个应用程序所示。最后，一节题为“补充”提供了一些后果，我们的结果近似的鞍点问题。
英文摘要：
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.
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