Journal Article

Numerical Analysis of Nonlinear Eigenvalue Problems

Chakir, Rachida, Maday, Yvon, Cancès, Eric

2010

Overview

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.

• Publisher: Springer US,Springer Nature B.V,Springer Verlag
• Subject: 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