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How statement and proof provenance work

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AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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A finite-dimensional BDDC/FETI-DP spectral match

Example

Let $\tilde{W} = R^{2}$ , $\tilde{S} = A = (\begin{array}{cc} 2 & 1 \\ 1 & 2 \end{array})$ , $\hat{W} = span (e_{1})$ , $R = e_{1}$ , $E_{0} = (1 0)$ , $B = (0 1)$ , and $B_{D}^{T} = e_{2}$ . Then $E_{0} R = 1$ , $B B_{D}^{T} = 1$ , and $R E_{0} + B_{D}^{T} B = I$ . The single BDDC and FETI-DP preconditioned eigenvalues both equal $4 / 3$ .

Facts & Assumptions

Given: The matrices and maps in the Example; the BDDC operator is $(E_{0} A^{- 1} E_{0}^{T}) (R^{T} A R)$ and the FETI-DP operator is $(B_{D} A B_{D}^{T}) (B A^{- 1} B^{T})$ .

[F1]

Coefficient-scaled averaging and BDDC: the BDDC preconditioned operator is formed from $E_{0}$ , $R$ and the partially assembled energy.

[F2]

FETI-DP dual system and preconditioner: the dual operator and preconditioner use $B$ and its scaled right inverse $B_{D}^{T}$ .

[F3]

BDDC and FETI-DP have matching nontrivial spectra: the two methods have matching positive nonunit eigenvalues.

Verification

1.1givenF1algebra

The determinant of $A$ is $3$ , so $A^{- 1} = \frac{1}{3} (\begin{array}{cc} 2 & - 1 \\ - 1 & 2 \end{array})$ . The BDDC coarse operator is $R^{T} A R = 2$ , while $E_{0} A^{- 1} E_{0}^{T} = 2 / 3$ ; their product is $4 / 3$ .

1.2givenF2algebra

The dual operator is $B A^{- 1} B^{T} = 2 / 3$ , and its preconditioner is $B_{D} A B_{D}^{T} = 2$ ; their product is also $4 / 3$ . Direct multiplication gives the three identities in the Example, so the abstract assumptions hold.

2.1givenF3step 1.1step 1.2∎

Both operators are one-dimensional, hence each has the single eigenvalue computed in steps 1.1 and 1.2. The common value is positive and different from $1$ , consistent with [F3].

Depends on

Used by

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Dependency tree · two levels

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Sources

Jan Mandel and Bedřich Sousedík, BDDC and FETI-DP under Minimalist Assumptions, §§3–4 (standard reference, not scraped)