A method for solving biharmonic-type equation with a singularly occuring small parameter
type: Item
instance of: scholarly article
title: A method for solving biharmonic-type equation with a singularly occuring small parameter
zbMATH Open document ID: 0779.35035
author: A. A. Abramov
author: V. I. Ul'yanova
published in: Computational Mathematics and Mathematical Physics
publication date: 1993-06-29T00:00:00Z
review text: An iterative algorithm to solve the boundary value problem \(- \varepsilon^ 2 \Delta^ 2 u+\Delta u=f\) in a domain \(\Omega\subset\mathbb{R}^ m\), \(u|_{\partial\Omega} ={{\partial u} \over {\partial n}}\Bigl|_{\partial \Omega}=0\) is proposed. For this purpose, the decomposition into two Poisson problems \[ - \varepsilon^ 2 \Delta u^{k+1}+ u^{k+1}=v^ k, \quad {{\partial u^{k+1}} \over {\partial n}} \biggl|_{\partial\Omega}=0, \quad \Delta v^ k=f, \quad v^ k|_{\partial\Omega} =v^ k_ 0, \] is employed with a special sequence \(v^ k_ 0\) to satisfy the boundary condition \(u|_{\partial\Omega}=0\) as \(k\to \infty\). The convergence is proved only if \(\Omega\) is a ball or parallelepiped. Some numerical tests are presented to demonstrate the computational efficiency of the algorithm.
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 65N12
zbMATH DE Number: 220262
zbMATH Keywords: iterative algorithm
zbMATH Keywords: boundary value problem
zbMATH Keywords: Poisson problems
zbMATH Keywords: convergence
zbMATH Keywords: ball
zbMATH Keywords: parallelepiped
zbMATH Keywords: computational efficiency
zbMATH Keywords: algorithm
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 220262