Dilum Upeksha Haththotuwa Gamage, A certified Helmholtz-defect Schur correction for Oseen-type incompressible-flow discretizations, Eur. J. Math. Appl. 6 (2026), Article ID 8.
Full Text: PDF
Volume 6 (2026), Article ID 8
https://doi.org/10.28919/ejma.2026.6.8
Published: 18/08/2026
Abstract:
Pressure Schur complements are the main algebraic bottleneck in mixed discretizations of incompressible Stokes and Oseen flow. Existing pressure-mass, pressure-Laplacian, pressure-convection-diffusion, and augmented-Lagrangian approximations can be effective, but they are often assessed by iteration counts rather than by an a posteriori statement that the pressure operator is reliable for the discrete incompressibility constraint. This paper proposes a certified Helmholtz-defect Schur correction for the symmetric energy Schur complement associated with Oseen-type systems. A baseline pressure operator is replaced, on an adaptively selected low-dimensional pressure subspace, by an exact Galerkin action of the energy Schur complement. The subspace is generated from algebraic divergence defects and optional randomized pressure probes, while acceptance is based on a deterministic spectral certificate. The construction is symmetric positive definite whenever the baseline pressure operator is symmetric positive definite, and it avoids the indefiniteness risk of purely additive low-rank corrections. We prove that a certificate below one implies spectral equivalence with the energy Schur complement, gives pressure-update error bounds, controls the Schur continuity response, and yields an explicit preconditioned conjugate-gradient convergence estimate. The method is intended as a theoretical reliability layer for computational-fluid-dynamics solvers: it does not claim to replace standard Schur approximations, but certifies and adaptively strengthens them in pressure directions detected by the discrete incompressibility residual. A reproducible manufactured spectral-grid experiment verifies the algebraic certificate, the pressure-error bound, and the predicted Krylov iteration behavior.
How to Cite:
Dilum Upeksha Haththotuwa Gamage, A certified Helmholtz-defect Schur correction for Oseen-type incompressible-flow discretizations, Eur. J. Math. Appl. 6 (2026), Article ID 8. https://doi.org/10.28919/ejma.2026.6.8