Numerical Analysis of Nonlinear Eigenvalue Problems

We provide a priori error estimates for variational approximations of the ground state energy, eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form −div( A ∇ u )+ Vu + f ( u 2 ) u = λ u , . We focus in particular on the Fourier spectral approximation (for periodic problem...

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Veröffentlicht in:Journal of scientific computing Jg. 45; H. 1-3; S. 90 - 117
Hauptverfasser: Cancès, Eric, Chakir, Rachida, Maday, Yvon
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Boston Springer US 01.10.2010
Springer Nature B.V
Springer Verlag
Schlagworte:
Algorithms
Approximation
Computational Mathematics and Numerical Analysis
Computer Science
Convergence
Eigenvalues
Eigenvectors
Energy of formation
Estimates
Ground state
Mathematical and Computational Engineering
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Nonlinearity
Numerical Analysis
Theoretical
Non linear eigenvalue problem
Finite element approximation
Numerical analysis
Ground state computations
Spectral and pseudo spectral approximations
ISSN:0885-7474, 1573-7691
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Zusammenfassung:We provide a priori error estimates for variational approximations of the ground state energy, eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form −div( A ∇ u )+ Vu + f ( u 2 ) u = λ u , . We focus in particular on the Fourier spectral approximation (for periodic problems) and on the ℙ 1 and ℙ 2 finite-element discretizations. Denoting by ( u δ , λ δ ) a variational approximation of the ground state eigenpair ( u , λ ), we are interested in the convergence rates of , , | λ δ − λ |, and the ground state energy, when the discretization parameter δ goes to zero. We prove in particular that if A , V and f satisfy certain conditions, | λ δ − λ | goes to zero as . We also show that under more restrictive assumptions on A , V and f , | λ δ − λ | converges to zero as , thus recovering a standard result for linear elliptic eigenvalue problems. For the latter analysis, we make use of estimates of the error u δ − u in negative Sobolev norms.
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ISSN:0885-7474
1573-7691
DOI:10.1007/s10915-010-9358-1