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Periodic midpoint collocation has an explicit local second-order error budget

Statement

Consider the augmented phase-fixed BVP of The phase-bordered periodic-orbit BVP residual on $[0, 1]$ , with exact solution $u_{*} \in C^{3}$ , field $\tilde{f}$ , and affine boundary map $B$ . Put $M_{2} = ∥ u_{*}^{''} ∥_{\infty}, M_{3} = ∥ u_{*}^{'''} ∥_{\infty}, L_{f} = \sup ∥ D \tilde{f} ∥_{2}, C_{0} = M_{3} / 24 + L_{f} M_{2} / 8.$ For every admitted mesh cell with midpoint $m_{i}$ , assume the segment joining $u_{*} (m_{i})$ to $(u_{*} (t_{i}) + u_{*} (t_{i + 1})) / 2$ lies in the field domain and $∥ D \tilde{f} ∥_{o p} \leq L_{f}$ on that segment. Let $G_{h}$ be the normalized midpoint residual on a uniform mesh and $Y^{*}$ the samples of $u_{*}$ . Suppose that for all $0 < h \leq h_{0}$ , $J_{h} = D G_{h} (Y^{*})$ is invertible with $∥ J_{h}^{- 1} ∥_{\infty \to \infty} \leq K$ , and on a fixed-radius ball of radius $r$ about $Y^{*}$ the maps $D G_{h}$ are $L_{G}$ -Lipschitz in the block maximum norm. Assume $2 K L_{G} r \leq 1 / 2$ , $K C_{0} h^{2} \leq r / 2$ , and $2 K^{2} L_{G} C_{0} h^{2} \leq 1 / 2$ . Then $G_{h}$ has a unique zero $Y_{h}$ in that ball. Every candidate $Y$ in the ball satisfying $∥ G_{h} (Y) ∥_{\infty} \leq ε$ obeys

∥ P_{h} Y - u_{*} ∥_{\infty} \leq 4 K ε + (2 K C_{0} + \frac{M_{2}}{8}) h^{2},

where $P_{h}$ is linear interpolation of the node values. Thus the residual and discretization constants are explicit in the supplied derivative and inverse bounds. For the augmented state, the estimate includes both the orbit coordinates and the period coordinate.

Facts & Assumptions

Given: The exact augmented solution, uniform mesh, normalized midpoint residual, bounds $M_{2}, M_{3}, L_{f}, K, L_{G}$ , and the stated radius and mesh inequalities.

[F1]

The midpoint residual has rows $(u_{i + 1} - u_{i}) / h - \tilde{f} ((u_{i} + u_{i + 1}) / 2)$ and the phase-periodic boundary row (Normalized implicit-midpoint collocation residual for a two-point BVP, Multiple-shooting and midpoint-collocation residuals for a phase-fixed periodic orbit).

[F2]

For a BVP with a boundedly invertible continuous linearization, its sampled midpoint Jacobian has a uniform inverse bound (A transverse phase border makes the periodic BVP linearization boundedly invertible, Uniform inverse bound for the sampled midpoint linearized BVP); here the explicit $K$ is assumed as a supplied upper bound for those discrete inverses.

[F3]

A contraction of a nonempty complete metric space into itself has a unique fixed point (A contraction of a nonempty complete metric space into itself has exactly one fixed point, the limit of the iterates from any starting point). The closed ball in the node space is complete: a block-maximum Cauchy sequence is Cauchy in each of its finitely many Euclidean blocks, each block converges by $R$ and $R^{n}$ for $n \geq 1$ with the Euclidean metric are complete, componentwise from the Cauchy criterion in $R$ , and the finite maximum of the block errors tends to zero; closedness keeps the limit in the ball.

[F4]

If $f : [a, b] \to R^{m}$ is differentiable with integrable $f^{'}$ then $\int_{a}^{b} f^{'} = f (b) - f (a)$ ; and a bounded derivative makes $f$ Lipschitz gives the componentwise integral of a continuous vector derivative. For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable bounds the Euclidean norm of that integral. Reversed integration limits use The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ . Apply these in each output block; taking the finite maximum preserves the same bound in the block maximum norm.

[F5]

If $∥ J^{- 1} (A - J) ∥ \leq 1 / 2$ , the perturbed operator is invertible with inverse norm at most $2 ∥ J^{- 1} ∥$ (Neumann series and small perturbations of bounded inverses).

[F6]

The chain rule differentiates a composition of totally differentiable maps (The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[F7]

The fundamental theorem of calculus applies coordinatewise to a continuously differentiable vector curve, and the norm of a Euclidean vector-valued integral is at most the integral of the norm (The second fundamental theorem: if $G$ is differentiable on $[a, b]$ with $G^{'} = f$ and $f$ is integrable, then $\int_{a}^{b} f = G (b) - G (a)$ , For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable).

Proof

1.1F1F4F6F7givenalgebra

On a mesh cell centered at $m_{i}$ , repeated vector FTC gives $u_{*}^{'} (m_{i} + s) = u_{*}^{'} (m_{i}) + s u_{*}^{''} (m_{i}) + r (s)$ , where $r (s) = \int_{0}^{s} \int_{0}^{v} u_{*}^{'''} (m_{i} + w) d w d v$ . For negative limits reverse their orientation before applying the norm inequality. Thus $∥ r (s) ∥_{2} \leq M_{3} ∣ s ∣^{2} / 2$ . Averaging over $[- h / 2, h / 2]$ cancels the odd term and gives a difference-quotient defect at most $M_{3} h^{2} / 24$ . Applying the same integral argument to $u_{*}$ at $m_{i} \pm h / 2$ bounds each first-order remainder by $M_{2} h^{2} / 8$ ; their average therefore bounds the endpoint-average displacement from $u_{*} (m_{i})$ by $M_{2} h^{2} / 8$ . Integrating $D \tilde{f}$ on the connecting segment bounds the corresponding field difference by $L_{f} M_{2} h^{2} / 8$ . The affine boundary residual is zero. Taking the maximum of the cell bounds gives $∥ G_{h} (Y^{*}) ∥_{\infty} \leq C_{0} h^{2}$ .

2.1F2F3F4F6F7step 1.1given

Define $T_{h} (Y) = Y - J_{h}^{- 1} G_{h} (Y)$ . The Jacobian Lipschitz bound gives $∥ T_{h} (Y) - T_{h} (Z) ∥_{\infty} \leq K L_{G} r ∥ Y - Z ∥_{\infty} \leq \frac{1}{4} ∥ Y - Z ∥_{\infty}$ on the ball. Also $∥ T_{h} (Y^{*}) - Y^{*} ∥_{\infty} \leq K C_{0} h^{2} \leq r / 2$ , so $T_{h}$ maps the ball into itself. By [F3], it has a unique fixed point $Y_{h}$ , and invertibility of $J_{h}$ makes fixed points exactly the zeros of $G_{h}$ . The fixed-point equation also gives $∥ Y_{h} - Y^{*} ∥_{\infty} \leq K C_{0} h^{2} / (1 - 1 / 4) \leq 2 K C_{0} h^{2}$ .

3.1F5step 2.1given

The perturbation bound $∥ D G_{h} (Y_{h}) - J_{h} ∥ \leq L_{G} ∥ Y_{h} - Y^{*} ∥$ implies $∥ J_{h}^{- 1} (D G_{h} (Y_{h}) - J_{h}) ∥ \leq 2 K^{2} L_{G} C_{0} h^{2} \leq 1 / 2$ . The small-perturbation inverse bound [F5] therefore gives $D G_{h} (Y_{h})^{- 1}$ with norm at most $2 K$ .

4.1F6F7step 3.1givenalgebra

Put $δ = Y - Y_{h}$ and define $R (t) = G_{h} (Y_{h} + t δ) - G_{h} (Y_{h}) - t D G_{h} (Y_{h}) δ$ for $0 \leq t \leq 1$ . The segment lies in the ball. By the chain rule [F6], $R' (t) = (D G_{h} (Y_{h} + t δ) - D G_{h} (Y_{h})) δ$ , so the Lipschitz bound gives $∥ R' (t) ∥_{\infty} \leq L_{G} t ∥ δ ∥_{\infty}^{2}$ . For each Euclidean block index $j$ , [F7] gives $∥ R_{j} (1) ∥_{2} \leq \int_{0}^{1} ∥ R_{j}' (t) ∥_{2} d t \leq \int_{0}^{1} ∥ R' (t) ∥_{\infty} d t$ ; taking the finite maximum over $j$ gives $∥ R (1) ∥_{\infty} \leq \int_{0}^{1} ∥ R' (t) ∥_{\infty} d t \leq (L_{G} / 2) ∥ δ ∥_{\infty}^{2}$ . Thus $G_{h} (Y) = D G_{h} (Y_{h}) (Y - Y_{h}) + R (1)$ with this remainder bound. Since both vectors lie in the radius- $r$ ball, $∥ Y - Y_{h} ∥_{\infty} \leq 2 r$ , and $2 K L_{G} r \leq 1 / 2$ yields $∥ Y - Y_{h} ∥_{\infty} \leq 2 K ε + \frac{1}{2} ∥ Y - Y_{h} ∥_{\infty}$ . Hence $∥ Y - Y_{h} ∥_{\infty} \leq 4 K ε$ .

5.1F4step 2.1step 4.1givenalgebra∎

Linear interpolation is a convex combination, so $∥ P_{h} (Y - Y_{h}) ∥_{\infty} \leq ∥ Y - Y_{h} ∥_{\infty}$ . On a cell $[a, b]$ of length $h$ , repeated FTC gives $u_{*} (t) - P_{h} Y^{*} (t) = - \int_{a}^{b} K_{t} (s) u_{*}^{''} (s) d s$ , where $K_{t} (s) = (b - t) (s - a) / h$ for $s \leq t$ and $K_{t} (s) = (t - a) (b - s) / h$ for $s \geq t$ . This kernel is nonnegative and has integral $(t - a) (b - t) / 2 \leq h^{2} / 8$ , so [F4] gives the interpolation bound $M_{2} h^{2} / 8$ . The sample error of $Y_{h}$ is at most $2 K C_{0} h^{2}$ by step 2.1. Combining these bounds with step 4.1 proves the displayed estimate.

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Sources

S. M. Rump, Verification methods: Rigorous results using floating-point arithmetic, Acta Numerica 19 (2010) (standard reference, not scraped)
E. J. Doedel and B. E. Oldeman, AUTO-07P full manual (2021) (standard reference, not scraped)