Approximation of the bifurcation function for elliptic boundary value problems

Abstract

The bifurcation function for an elliptic boundary value problem is a vector field B(ω) on R d whose zeros are in a one-to-one correspondence with the solutions of the boundary value problem. Finite element approximations of the boundary value problem are shown to give rise to an approximate bifurcation function B h (ω), which is also a vector field on R d . Estimates of the difference B(ω) − B h (ω) are derived, and methods for computing B h (ω) are discussed.

Computation of Morse decompositions for semilinear elliptic PDEs

“…Since E h (λ) → E(λ) as h → 0, it follows that we also have E h (λ) → E(λ) as h → 0. The next result, proven in [19], gives an estimate of the difference between the computable quantities σ h (ω, λ), z h (ω, λ), B h (ω, λ) and the theoretical quantities σ(ω, λ)z(ω, λ), B(ω, λ).…”

“…This article follows and extends the methods and results of our previous article [19], which discusses the basic process of reduction to a bifurcation function B(ω, λ) and the use of finite elements to obtain an approximating bifurcation function B h (ω, λ). The main results of [19] are error estimates on the approximations B h ≈ B and z h ≈ z, where z and z h are potentially (see Section II) a solution and its approximation. A key ingredient in these estimates is an error estimate for a solution v = σ(ω, λ) of an auxiliary boundary value problem.…”

“…Its solution requires the use of projections and a pseudo-inverse. Implementation and algorithmic issues are discussed in [19].…”

“…These estimates, which show the Morse indices and energy levels can be accurately computed, are presented in Section IV. The article concludes with several examples in Section V. Remarks on implementation and examples discussing convergence can be found in [19].…”

Fibers and global geometry of functions

“…In particular, one may search for folds in nonlinear maps defined on functions with unbounded domains, which are natural in physical situations. Fibers also provide the conceptual starting point for algorithms that solve a class of partial differential equations, an idea originally suggested by Smiley ([31], [32]) and later implemented for finite spectral interaction of the Dirichlet Laplacian on rectangles in [8].…”