Robustness of pressure-coupled additive CAV relaxation

Establish the robustness of the candidate pressure-coupled additive CAV relaxation for the indefinite velocity–pressure immersed-boundary saddle-point system.

Background

The paper develops CAV as a multiplicative Schwarz smoother for the reduced Eulerian velocity–pressure systems arising from semi-implicit immersed boundary discretizations. Its appendix derives an additive Schwarz model for the elasticity-augmented momentum block and argues that coupling-aware patches can capture elastic near-kernel components while maintaining bounded interaction and stable-decomposition constants.

The appendix then proposes a pressure-coupled additive CAV operator using weighted restrictions over the full velocity–pressure patches. Although a simple reciprocal patch-count weighting satisfies a partition-of-unity condition and restricted additive injection is identified as another option, the effects of damping, weighting, and the indefinite pressure block are not analyzed. The robustness of this additive formulation therefore remains unresolved.

References

The heuristic momentum-block model motivates the patch family and suggests which near-kernel components it should capture, but damping, weighting, and the indefinite pressure block require additional analysis. Equation~eq:additive_cav_saddle therefore provides a concrete candidate for pressure-coupled additive CAV relaxation; its robustness for the indefinite system remains to be established.

— Coupling-Aware Vanka Smoothing for Multigrid Preconditioning of the Implicit Immersed Boundary Equations  (2608.14310 - Gruninger et al., 14 Aug 2026) in Appendix, Section “Additive Schwarz motivation for CAV patch selection,” immediately following Equation (additive_cav_saddle)