2017/06/27 by Mario Amrein, Amrein, Mario
Engineering · Mathematics · #Advanced Numerical Methods in Computational Mathematics #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.1706.09299
openalex publication_date 2017/06/27 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
In this paper we study the behavior of finite dimensional fixed point\niterations, induced by discretization of a continuous fixed point iteration\ndefined within a Banach space setting. We show that the difference between the\ndiscrete sequence and its continuous analogue can be bounded in terms depending\non the mesh size of the discretization and the contraction factor, defined by\nthe continuous iteration. Furthermore, we show that the comparison between the\nfinite dimensional and the continuous fixed point iteration naturally paves the\nway towards a General a posteriori error analysis that can be used within the\nframework of a fully adaptive solution procedure. In order to demonstrate our\napproach, we use the Galerkin approximation of singularly perturbed semilinear\nmonotone problems. Our scheme combines the fixed point iteration with an\nadaptive finite element discretization procedure (based on a robust a\nposteriori error analysis), thereby leading to a fully adaptive\nFixed-Point-Galerkin scheme. Numerical experiments underline the robustness and\nreliability of the proposed approach.\n