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Numerical analysis of nonlinear eigenvalue problems
Eric Cancès, Rachida Chakir, Yvon Maday
TL;DR
The paper derives a priori error estimates for variational approximations of nonlinear ground-state eigenpairs. It develops general and discretization-specific analyses, including finite-element spaces, and proves quadratic eigenvalue-error convergence under the analyzed conditions.
Problem
The paper addresses the derivation of a priori error estimates for variational approximations of the ground state eigenpair.
Method
The analysis establishes general error bounds and then treats specific discretizations, including finite-element spaces constructed from meshes and polynomial element spaces.
Results
The eigenvalue error |λδ−λ| converges at least as ∥uδ−u∥^2 under the stated estimates.
Takeaways & Limitations
The paper provides error bounds for eigenvalue, eigenvector, and energy approximations in a general framework and for more specific discretizations.
Takeaways & Limitations
Numerical integration can dramatically pollute eigenvalue approximations because it removes the better negative-norm estimates underlying doubled convergence.
Abstract
from arXiv · showhide
We provide a priori error estimates for variational approximations of the ground state eigenvalue and eigenvector of nonlinear elliptic eigenvalue problems of the form $-{div} (A\nabla u) + Vu + f(u^2) u = λu$, $\|u\|_{L^2}=1$. We focus in particular on the Fourier spectral approximation (for periodic problems) and on the $¶_1$ and $¶_2$ 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 $\|u_δ-u\|_{H^1}$, $\|u_δ-u\|_{L^2}$ and $|λ_δ-λ|$, when the discretization parameter $δ$ goes to zero. We prove that if $A$, $V$ and $f$ satisfy certain conditions, $|λ_δ-λ|$ goes to zero as $\|u_δ-u\|_{H^1}^2+\|u_δ-u\|_{L^2}$. We also show that under more restrictive assumptions on $A$, $V$ and $f$, $|λ_δ-λ|$ converges to zero as $\|u_δ-u\|_{H^1}^2$, thus recovering a standard result for {\em linear} elliptic eigenvalue problems. For the latter analysis, we make use of estimates of the error $u_δ-u$ in negative Sobolev norms.
1 Introduction
The paper develops a priori error estimates for variational approximations of ground-state eigenpairs in nonlinear elliptic eigenvalue problems, covering general discretizations, Fourier modes, and P1/P2 finite elements. Under stated assumptions, it establishes convergence results for eigenvectors, eigenvalues, and energies, including improved eigenvalue estimates based on negative Sobolev norms.
- Nonlinear eigenvalue problems arise in models including nonlinear elasticity, Bose–Einstein condensates, and electronic-structure calculations.
- The analysis assumes ellipticity and symmetry of A together with regularity, convexity, growth, and continuity conditions on the nonlinear energy F.Additional restrictions on the exponents are introduced for sharper estimates.
- The continuous problem has a unique positive solution up to sign, whereas discrete minimizers need not be unique because the relevant discrete set is generally nonconvex.
- The paper derives a priori error estimates for variational approximations of the ground-state eigenpair over finite-dimensional spaces Xδ.The discrete minimizer is selected with nonnegative L2 inner product against the exact positive solution.
- The eigenvalue error converges at least as ∥uδ−u∥H1^2 plus a term controlled by ∥uδ−u∥L6/(5−2q).The first contribution is nonnegative and vanishes with the squared H1 error.
- The study covers Fourier spectral discretization for periodic problems and P1/P2 finite-element discretizations, while also discussing numerical integration.
2 Basic error analysis
The paper develops a general variational framework for bounding eigenvector, eigenvalue, and energy errors under stated assumptions. The analysis uses coercivity, adjoint problems, and negative Sobolev estimates to sharpen eigenvalue bounds.
- The general framework estimates H^1, L^2, eigenvalue, and energy errors for discretized ground states.
- Adjoint problems on the orthogonal complement of u support L^2 and negative-norm error estimates.
- Theorem 1 provides the main error bounds under assumptions (2)-(6) and (13), with refinements under assumptions (7) and (8).
- The H^1 eigenvector error is controlled by the best-approximation error J_δ in H^1.
- Negative Sobolev estimates improve eigenvalue bounds toward the quadratic form familiar from linear elliptic eigenvalue problems.
3 Fourier expansion
For periodic problems, the paper analyzes Fourier spectral Galerkin approximations and derives regularity-dependent convergence estimates. Numerical tests show algebraic decay rates consistent with the theoretical bounds.
- Theorem 2 establishes convergence and error bounds under Sobolev regularity of the potential and smoothness assumptions on F.
- The Fourier approximation truncates the periodic solution to a finite-dimensional space of Fourier modes.
- Under additional regularity, eigenvalue convergence is twice as fast as H^1 eigenvector convergence, as in the linear problem.
- In the numerical test, H^1, L^2, H^-1, and eigenvalue errors decay respectively as N^-2.67, N^-3.67, N^-4.67, and N^-5.
- The observed rates agree with the theoretical rates N^-2.5+ε, N^-3.5+ε, N^-4.5+ε, and N^-5+ε.
4 Finite element discretization
For nonperiodic problems, the paper studies P1 and P2 finite-element spaces on regular triangulations and derives a priori error estimates under regularity assumptions. The estimates are supported by numerical tests.
- Finite-element spaces use continuous piecewise polynomials of degree k on regular triangulations, with discretization parameter h.
- Theorem 3 gives P1 and P2 finite-element error estimates under assumptions on V, F, and the mesh.
- For P1 elements, the H^1 and L^2 eigenvector errors are bounded by Ch and Ch^2, respectively.
- Additional regularity of V and F enables sharper P2 estimates through elliptic regularity and adjoint-problem arguments.
- Numerical results show good agreement for P1 and excellent agreement for P2 with the theoretical error estimates.
5 The effect of numerical integration
The section examines how numerical integration affects Fourier and finite-element approximations, separating discretization error from integration error and testing their convergence behavior.
- Setup: Numerical integration is introduced as a practical issue in evaluating nonlinear terms for finite-element and Fourier discretizations.Finite elements use quadrature, while Fourier methods evaluate relevant coefficients with FFTs on an integration grid that may differ from the discretization grid.
- Fourier implementation: For periodic problems with F(t)=ct^2, an FFT grid twice as fine as the discretization grid computes the nonlinear term exactly up to round-off errors.This exactness relies on the specific quadratic nonlinearity considered.
- Error bounds: The error bounds decompose into discretization error and numerical integration error for the H1, L2, and negative-norm estimates.The first component measures ∥uN−u∥H1, while the second measures ∥uN,Ng−uN∥H1; analogous interpretations apply to the other bounds.
- Limitation: Numerical integration can severely pollute eigenvalue convergence because negative-norm interpolation errors generally have only the same order as L2 errors.The sharper negative-norm estimates underpinning doubled eigenvalue convergence are therefore lost.
- Numerical behavior: The eigenvalue-error curve can be non-monotone because λN,Ng−λ may be positive or negative as N and Ng vary.This sign variation explains why increasing the discretization dimension does not necessarily produce a monotone absolute-error curve.
- Numerical behavior: For fixed N, smaller Ng values make numerical integration errors dominate, and the errors decay linearly with log10 Ng at slope close to −2.The observed behavior concerns H1, L2, H−1, and eigenvalue errors.
6 Appendix: properties of the ground state
The appendix establishes the variational and spectral properties of the ground state, including existence, positivity, uniqueness up to sign, and simplicity of the associated lowest eigenvalue.
- Assumptions: The assumptions include d=1, 2, or 3, uniform boundedness and coercivity of A, and V∈Lq(Ω) with q>max(1,d/2).These conditions support boundedness below, compactness, and existence of minimizers.
- Variational structure: The minimization problems admit a unique nonnegative density minimizer and exactly two wave-function minimizers, u and −u.The relationship follows from the change of variables ρ=v^2 and strict convexity of the density functional.
- Ground-state regularity: The ground-state function u is positive and Hölder continuous, with u∈C0,α(Ω) for some 0<α<1.The positivity conclusion uses elliptic regularity and the Harnack inequality.
- Spectral characterization: The associated λ is the lowest eigenvalue of Au and is non-degenerate, hence simple.The proof uses compact resolvent, the min-max principle, positivity of the ground-state eigenfunction, and the resulting nonzero overlap with u.
- Ground-state characterization: Every other normalized nonlinear eigenpair has eigenvalue strictly above λ, unless its eigenvector is ±u and its eigenvalue equals λ.Thus the ground state is isolated among normalized solutions up to the sign symmetry.