In this paper we study the behaviour of finite dimensional fixed point iterations, induced by discretization of a continuous fixed point iteration defined within a Banach space setting. We show that the difference between the discrete sequence and its continuous analogue can be bounded in terms depending on the discretization of the infinite dimensional space and the contraction factor, defined by the continuous iteration. Furthermore, we show that the comparison between the finite dimensional and the continuous fixed point iteration naturally paves the way towards a general a posteriori error analysis that can be used within the framework of a fully adaptive solution procedure. In order to demonstrate our approach, we use the Galerkin approximation of singularly perturbed semilinear monotone problems. Our scheme combines the fixed point iteration with an adaptive finite element discretization procedure (based on a robust a posteriori error analysis), thereby leading to a fully adaptive fixed-point-Galerkin scheme. Numerical experiments underline the robustness and reliability of the proposed approach.

中文翻译：

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半线性椭圆型偏微分方程的自适应不动点迭代

在本文中，我们研究了有限维固定点迭代的行为，该行为是由Banach空间设置内定义的连续固定点迭代的离散化引起的。我们表明，离散序列与其连续类似物之间的差异可以根据无限维空间的离散化和由连续迭代定义的收缩因子来限制。此外，我们表明，有限维和连续定点迭代之间的比较自然地为一般后验误差分析铺平了道路，后者可以在完全自适应的求解过程的框架内使用。为了证明我们的方法，我们使用奇摄动半线性单调问题的Galerkin逼近。我们的方案将定点迭代与自适应有限元离散化程序（基于鲁棒的后验误差分析）相结合，从而导致了完全自适应的定点-Galerkin方案。数值实验强调了该方法的鲁棒性和可靠性。