FINITE ELEMENT APPROXIMATION OF INVARIANT MANIFOLDS BY THE PARAMETERIZATION METHOD

The paper formulates the invariance equation F(P(θ)) = DP(θ) Λ θ for local unstable manifolds and, after expanding P as a multivariate power series, derives linear homological equations whose right-hand sides depend only on lower-order coefficients. In the PDE setting, these become linear elliptic problems which (in weak/finite-element form) are solved recursively (see the invariance equation and push-forward expansion, and the FEM homological equations and their recursive solution). The candidate solution reproduces exactly this coefficient-by-coefficient derivation in operator form, adds the standard non-resonance condition n·λ ∉ spec(DF(u0)) to guarantee invertibility of each elliptic resolvent, and emphasizes that the same boundary conditions are inherited by the coefficient problems. Up to this explicit non-resonance hypothesis—implicit in the paper’s FEM linear solves—the arguments align closely. See the paper’s Eq. (2.5) and series push-forward DP(θ)Λθ = Σ(n·λ) p_n θ^n, and the weak-form homological systems solved recursively for concrete PDEs .