Numerical analysis of semilinear elliptic equations with finite spectral interaction
type: Item
instance of: scholarly article
title: Numerical analysis of semilinear elliptic equations with finite spectral interaction
zbMATH Open document ID: 1252.65187
author: José Teixeira Cal Neto
author: Carlos Tomei
published in: Journal of Mathematical Analysis and Applications
publication date: 2012-08-08T00:00:00Z
full work available at URL: https://arxiv.org/abs/1107.5783
review text: The authors explore numerically a global Lyapunov-Schmidt decomposition to obtain a numerical algorithm for solving \(-\Delta u(x)-f(x,u(x))=g(x)\) with Dirichlet boundary conditions in a bounded domain \(\Omega .\) The nonlinearities are non-resonant and have finite spectral interaction: no eigenvalue of \(-\Delta _{D}\) is an endpoint of \(\overline{\partial _{2}f\left( \Omega ,\mathbb{R}\right) }\), which in turn only contains a finite number of eigenvalues (\( \partial _{2}\) is the partial derivative with respect to the second variable). The authors use a class of \(C^{1}\) maps \(F:X\rightarrow Y\) between Banach spaces. Split \(X=W_{X}\oplus V_{X}\) and \(Y=W_{Y}\oplus V_{y}\) into closed horizontal and vertical subspaces. Define complementary projections \(P_{Y},Q_{Y}:Y\rightarrow Y\) such that \(\mathrm {Ran}P_{Y}=W_{Y}\), \(\mathrm {Ran} Q_{Y}=V_{Y}.\) The bifurcation equations related to the decomposition for \( F\left( x\right) =y\) are \(P_{Y}F(w+v)=P_{Y}y,Q_{Y}F(w+v)=Q_{Y}y\), \(w\in W_{X}\), \(v\in V_{X}\). The algorithm first computes \(w\) using the finite element method. The search for solutions of \(F\left( x\right) =y\) then reduces to inverting a computable map between isomorphic finite dimensional subspaces \( V_{X}\) and \(V_{Y}.\)
reviewed by: Adrian Carabineanu
Mathematics Subject Classification ID: 65N30
Mathematics Subject Classification ID: 35J61
zbMATH DE Number: 6064283
zbMATH Keywords: semilinear elliptic equations
zbMATH Keywords: finite element methods
zbMATH Keywords: Lyapunov-Schmidt decomposition
zbMATH Keywords: algorithm
MaRDI profile type: Publication
OpenAlex ID: W2964173318
arXiv ID: 1107.5783
cites work: Q3749777
cites work: A mountain pass method for the numerical solution of semilinear elliptic problems
cites work: Multiple solutions for a semilinear boundary value problem: a computational multiplicity proof.
cites work: Computer-assisted proofs for semilinear elliptic boundary value problems
cites work: Nonlinear Integral Equations of the Hammerstein Type
cites work: On the inversion of some differentiable mappings with singularities between Banach spaces
cites work: Q5184084
cites work: Q4095271
cites work: Q3669825
cites work: A finite element method for computing the bifurcation function for semilinear elliptic BVPs
cites work: A principle of reduced stability for reaction-diffusion equations
cites work: Approximation of the bifurcation function for elliptic boundary value problems
cites work: An algorithm for finding all solutions of a nonlinear system
cites work: Computation of Morse decompositions for semilinear elliptic PDEs
cites work: Q4123888
cites work: Morin singularities and global geometry in a class of ordinary differential operators
cites work: Q4764181
cites work: Applications of global analysis to specific nonlinear eigenvalue problems
DOI: 10.1016/J.JMAA.2012.04.073
Recommended article: Q4938347
Recommended article: Q3337418
Recommended article: Q5352012
Recommended article: Q5503186
Recommended article: Numerical computations for singular semilinear elliptic boundary value problems
Recommended article: Q4504857
Recommended article: On the numerical solution of some semilinear elliptic problems
Recommended article: Q5687333
Recommended article: Numerical Methods for Semilinear Parabolic Equations
Recommended article: Numerical analysis of nonlinear elliptic-parabolic equations
description: scientific article; zbMATH DE number 6064283