Motivation
Uniform partition-of-unity weighting on slope-carrying subdomains is a safe stopgap, not the theoretically correct fix. Prior investigation into the same convergence hazard established that multiplicity-based 1/√c weighting is wrong because it is blind to whether a shared DOF is a stiff connector or redundant bulk, and derived the correct two-sided weighting as inverse-diagonal (resolvability): w_i[j]² = (A_i⁺)_jj / Σ_k (A_k⁺)_kj. That weighting matched or beat uniform weighting everywhere tested — identical on weakly-connected designs, ~6–13% fewer iterations on well-connected slope designs — but was never implemented: no cheap faithful shortcut was found (the local Schur diagonal fails on chain topologies since connector information lives in its off-diagonals; Hutchinson probing of the local pseudoinverse is too high-variance on the same chains). A faithful implementation needs exact selected inversion — e.g. Takahashi diag(A⁻¹) over the approx-Cholesky factor, or one reduced solve per shared DOF.
Goal
Determine whether a faithful (or acceptably approximate) inverse-diagonal partition-of-unity weighting can be computed at production scale for slope-carrying subdomains, and if so, implement it in place of the uniform-weight stopgap.
Motivation
Uniform partition-of-unity weighting on slope-carrying subdomains is a safe stopgap, not the theoretically correct fix. Prior investigation into the same convergence hazard established that multiplicity-based
1/√cweighting is wrong because it is blind to whether a shared DOF is a stiff connector or redundant bulk, and derived the correct two-sided weighting as inverse-diagonal (resolvability):w_i[j]² = (A_i⁺)_jj / Σ_k (A_k⁺)_kj. That weighting matched or beat uniform weighting everywhere tested — identical on weakly-connected designs, ~6–13% fewer iterations on well-connected slope designs — but was never implemented: no cheap faithful shortcut was found (the local Schur diagonal fails on chain topologies since connector information lives in its off-diagonals; Hutchinson probing of the local pseudoinverse is too high-variance on the same chains). A faithful implementation needs exact selected inversion — e.g. Takahashidiag(A⁻¹)over the approx-Cholesky factor, or one reduced solve per shared DOF.Goal
Determine whether a faithful (or acceptably approximate) inverse-diagonal partition-of-unity weighting can be computed at production scale for slope-carrying subdomains, and if so, implement it in place of the uniform-weight stopgap.