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Discrete harmonic extension minimizes energy

Statement

For every admissible local trace $g_{i}$ , $H_{i} g_{i}$ from Discrete harmonic extension and interface Schur complement is the unique finite-element function with trace $g_{i}$ minimizing $a_{i} (v, v)$ . The minimum is $g_{i}^{T} S_{i} g_{i}$ .

Facts & Assumptions

Given: The local block matrix $K_{i}$ , positive definite interior block $K_{I I}^{(i)}$ , trace $g_{i}$ , harmonic lift $H_{i} g_{i}$ , and Schur matrix $S_{i}$ .

[F1]

Discrete harmonic extension and interface Schur complement: $H_{i} g_{i} = (- K_{I I}^{- 1} K_{I Γ} g_{i}, g_{i})^{T}$ and $S_{i} = K_{Γ Γ} - K_{Γ I} K_{I I}^{- 1} K_{I Γ}$ .

[F2]

Floating local kernels and positive interior blocks: $K_{I I}^{(i)}$ is positive definite, so its quadratic form is nonnegative and vanishes only at zero.

Proof

1.1givenF1algebra

Any local vector with trace $g_{i}$ has the form $v = H_{i} g_{i} + (z, 0)^{T}$ . By [F1], the interior block row of $K_{i} H_{i} g_{i}$ is $K_{I I} (- K_{I I}^{- 1} K_{I Γ} g_{i}) + K_{I Γ} g_{i} = 0$ . Hence the energy cross term between $H_{i} g_{i}$ and $(z, 0)$ vanishes, and $v^{T} K_{i} v = (H_{i} g_{i})^{T} K_{i} (H_{i} g_{i}) + z^{T} K_{I I} z$ .

2.1givenF1F2step 1.1algebra∎

By [F2], the last term is nonnegative and is zero only for $z = 0$ . Thus $H_{i} g_{i}$ is the unique minimizer. Multiplying the block expression in [F1] gives $(H_{i} g_{i})^{T} K_{i} (H_{i} g_{i}) = g_{i}^{T} (K_{Γ Γ} - K_{Γ I} K_{I I}^{- 1} K_{I Γ}) g_{i} = g_{i}^{T} S_{i} g_{i}$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Olof B. Widlund, Introduction to the Design and Theory of Domain Decomposition Algorithms, DD20 Tutorial, pp. 35–37 (standard reference, not scraped)
Oliver Rheinbach, Parallel Scalable Iterative Substructuring, Chapter 2, §§2.1–2.3 (standard reference, not scraped)