Numerical verification of solutions for nonlinear elliptic problems using an \(L^\infty\) residual method
type: Item
instance of: scholarly article
title: Numerical verification of solutions for nonlinear elliptic problems using an \(L^\infty\) residual method
zbMATH Open document ID: 0893.35006
author: Mitsuhiro T. Nakao
author: Nobito Yamamoto
published in: Journal of Mathematical Analysis and Applications
publication date: 1998-02-09T00:00:00Z
review text: The authors propose an ingenious numerical enclusion method with \(L^\infty\) error bounds for the solutions of the boundary value problem \(-\Delta u=f(u)\) in a convex polygonal subdomain \(\Omega\subset \mathbb{R}^2\) with \(u=0\) on \(\partial\Omega\). The method needs not the assumption of the existence of the solution, but this follows as a result from the estimates. As the main tool, a constructive error estimate for the finite element approximation of the Poisson equation \(-\Delta v=g\) \((g\in L^2(\Omega))\), \(v=0\) on \(\partial \Omega\) is employed. The final results are demonstrated for the equation \(-\Delta u=u^2\).
Mathematics Subject Classification ID: 65M99
Mathematics Subject Classification ID: 65N99
Mathematics Subject Classification ID: 65M15
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 1115469
zbMATH Keywords: enclosure method
zbMATH Keywords: existence
zbMATH Keywords: constructive error estimate
reviewed by: Jan Chrastina
describes a project that uses: PROFIL/BIAS
MaRDI profile type: Publication
full work available at URL: jmaa.1997.5712
OpenAlex ID: W2004220260
cites work: Q3345690
cites work: A‐posteriori error estimates for the finite element method
cites work: Q4840766
cites work: Q5733282
cites work: A numerical approach to the proof of existence of solutions for elliptic problems
cites work: A numerical verification method for the existence of weak solutions for nonlinear boundary value problems
cites work: Q4860584
cites work: Q4284594
cites work: Pointwise a Posteriori Error Estimates for Elliptic Problems on Highly Graded Meshes
cites work: Q3985896
cites work: Explicit \(H_ 2\)-estimates and pointwise bounds for solutions of second- order elliptic boundary value problems
cites work: Q4321536
cites work: Q4100478
cites work: A Posteriori Error Estimates for Nonlinear Problems. Finite Element Discretizations of Elliptic Equations
cites work: Numerical verifications of solutions for nonlinear elliptic equations
cites work: Numerical verifications for solutions to elliptic equations using residual iterations with a higher order finite element
cites work: Verified computations of solutions for nondifferentiable elliptic equations related to MHD equilibria
DOI: 10.1006/JMAA.1997.5712
description: scientific article; zbMATH DE number 1115469