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The stable-splitting guarantee worsens as overlap shrinks

Example

Assume Countable Choice. For any admissible overlap radius $δ$ in the structured family of Stable two-level P1 decomposition for structured Poisson meshes, put $r = H / δ$ . Substitution into the proved stable-decomposition guarantee gives $C_{0} \leq C_{F E} (1 + r^{2}) .$ Along admissible families with $δ / H \to 0$ , this upper-bound formula grows. This substitution does not assert that the actual stable constant or condition number must diverge.

Facts & Assumptions

Given: Countable Choice and a member of the structured P1 Poisson family of Stable two-level P1 decomposition for structured Poisson meshes and its admissible overlap radius $δ$ .

[F1]

The theorem proves $C_{0} \leq C_{F E} (1 + (H / δ)^{2})$ , with $C_{F E}$ independent of $h$ , $H$ , and $δ$ (Stable two-level P1 decomposition for structured Poisson meshes).

Verification

1.1F1givenalgebra

Put $r = H / δ$ . Substituting into [F1] gives $C_{0} \leq C_{F E} (1 + r^{2})$ .

2.1step 1.1givenalgebra∎

To exhibit admissible radii with $δ / H \to 0$ , fix $M, d$ , put $a = 1 / M$ , choose $q_{0} = ⌈ \sqrt{d} ⌉$ , and take $δ = q_{0} a / n$ as $n \to \infty$ with $n \geq q_{0}$ . Then $h = \sqrt{d} a / n \leq δ \leq a$ , and $r = H / δ = \sqrt{d} n / q_{0} \to \infty$ . The right side of the one-sided upper-bound formula grows; it supplies no lower estimate on the actual stable constant or condition number.

Depends on

Stable two-level P1 decomposition for structured Poisson meshes

Used by

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Dependency tree · two levels

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Sources

Jinchao Xu, Iterative Methods by Space Decomposition and Subspace Correction, SIAM Review 34(4) (1992) (standard reference, not scraped)
Peter Bastian, Lecture Notes on Parallel Solution of Large Sparse Linear Systems, University of Heidelberg (2015) (standard reference, not scraped)