A modified iterated projection method adapted to a nonlinear integral equation
| Autor: | Laurence Grammont, Mario Ahues, Paulo B. Vasconcelos |
|---|---|
| Přispěvatelé: | Institut Camille Jordan [Villeurbanne] (ICJ), École Centrale de Lyon (ECL), Université de Lyon-Université de Lyon-Université Claude Bernard Lyon 1 (UCBL), Université de Lyon-Université Jean Monnet [Saint-Étienne] (UJM)-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Centre National de la Recherche Scientifique (CNRS), Université de Lyon-Institut National des Sciences Appliquées de Lyon (INSA Lyon), Université de Lyon-Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-Université Jean Monnet [Saint-Étienne] (UJM)-Centre National de la Recherche Scientifique (CNRS) |
| Jazyk: | angličtina |
| Rok vydání: | 2016 |
| Předmět: |
Discretization
Applied Mathematics Mathematical analysis Linear system MathematicsofComputing_NUMERICALANALYSIS 010103 numerical & computational mathematics Fredholm integral equation Superconvergence 01 natural sciences Integral equation Local convergence 010101 applied mathematics Computational Mathematics Nonlinear system symbols.namesake symbols 0101 mathematics Newton's method ComputingMilieux_MISCELLANEOUS [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA] Mathematics |
| Zdroj: | Applied Mathematics and Computation Applied Mathematics and Computation, Elsevier, 2016, 276, pp.432-441. ⟨10.1016/j.amc.2015.12.019⟩ |
| ISSN: | 0096-3003 |
| DOI: | 10.1016/j.amc.2015.12.019⟩ |
| Popis: | The classical way to tackle a nonlinear Fredholm integral equation of the second kind is to adapt the discretization scheme from the linear case. The Iterated projection method is a popular method since it shows, in most cases, superconvergence and it is easy to implement. The problem is that the accuracy of the approximation is limited by the mesh size discretization. Better approximations can only be achieved for fine discretizations and the size of the linear system to be solved then becomes very large: its dimension grows up with an order proportional to the square of the mesh size. In order to overcome this difficulty, we propose a novel approach to first linearize the nonlinear equation by a Newton-type method and only then to apply the Iterated projection method to each of the linear equations issued from the Newton method. We prove that, for any value (large enough) of the discretization parameter, the approximation tends to the exact solution when the number of Newton iterations tends to infinity, so that we can attain any desired accuracy. Numerical experiments confirm this theoretical result. |
| Databáze: | OpenAIRE |
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