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Local Newton convergence on the selected midpoint collocation branch

Statement

For each sufficiently fine mesh, let $Y_{h}$ be the local root of the normalized midpoint residual supplied by Local second-order midpoint collocation error with an algebraic residual budget. On a sufficiently small ball about $Y_{h}$ , the exact-Jacobian finite-budget Newton procedure in Finite-budget implicit-midpoint collocation for a nonlinear BVP has a basin in which every attempted update is valid, remains in the ball, and satisfies a quadratic error estimate. The radius may depend on $h$ ; a finite budget may stop early, and arbitrary initial node vectors receive no convergence guarantee.

Facts & Assumptions

Given: A sufficiently fine fixed mesh, the local root $Y_{h}$ , the C2 data and conditioning assumptions of the midpoint branch theorem, tolerance $τ \geq 0$ , budget $M \in N_{0}$ , and an initial vector in the local Newton ball.

[F1]

Local second-order midpoint collocation error with an algebraic residual budget: $G_{h} (Y_{h}) = 0$ and $D G_{h} (Y_{h})$ is invertible; the theorem defines $C^{2}$ as continuously differentiable with continuous second total derivatives.

[F2]

Normalized implicit-midpoint collocation residual for a two-point BVP: $G_{h}$ is the normalized square collocation residual.

[F3]

Finite-budget implicit-midpoint collocation for a nonlinear BVP: the procedure applies the finite exact-Jacobian Newton method to $G_{h}$ on its open nodal domain.

[F4]

Updated-Jacobian Newton has local quadratic convergence at a simple root: a C2 square map with an invertible derivative at a root and locally Lipschitz Jacobian has a local quadratic Newton basin.

Proof

1.1givenF1F2algebra

By [F1], the branch theorem’s $C^{2}$ convention means continuous second total derivatives. For this fixed mesh, $G_{h}$ maps the open nodal domain in $R^{d (N + 1)}$ to the same-dimensional residual space. Since $f$ and $B$ are $C^{2}$ , composition with the affine midpoint maps shows that $G_{h}$ is $C^{2}$ ; on a closed ball around $Y_{h}$ inside the domain its continuous second derivatives are bounded, so its Jacobian is Lipschitz there.

2.1givenF1F3F4step 1.1∎

Fact [F1] gives the zero and invertible derivative required by [F4]. Applying [F4] to this fixed finite-dimensional map yields a positive local radius and quadratic estimate. By [F3] the prescribed algorithm follows those exact Newton updates while the iterates remain in the ball; its finite budget can stop early and imposes no assertion for starts outside the basin.

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Sources

E. Hairer and G. Wanner, Solving Ordinary Differential Equations II, Chapter VI: Iterative Methods—Nonlinear Equations (standard reference, not scraped)