自适应固定点迭代法用于半线性椭圆偏微分方程
Adaptive Fixed Point Iterations for Semilinear Elliptic Partial Differential Equations
AI总结：
本文研究了有限维固定点迭代在Banach空间设定下的行为，通过网格大小和收缩因子界定了离散序列与连续解的差异，并展示了自适应有限元离散化方法在半线性单调问题中的应用。
摘要：
本文研究了在Banach空间设定下，由连续固定点迭代离散化诱导出的有限维固定点迭代的行为。我们证明了离散序列与其连续类比之间的差异可以以依赖于网格大小和由连续迭代定义的收缩因子的项来界。此外，我们展示了有限维和连续固定点迭代之间的比较自然地为后验误差分析铺平了道路，可用于完全自适应求解过程的框架中。为了展示我们的方法，我们使用Galerkin逼近来处理奇异扰动的半线性单调问题。我们的方案结合了固定点迭代与自适应有限元离散化过程（基于稳健的后验误差分析），从而得到完全自适应的Fixed-Point-Galerkin方案。数值实验验证了所提出方法的鲁棒性和可靠性。
英文摘要：
In this paper we study the behavior 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 mesh size of the discretization 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.
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