Block Preconditioning for Shifted Boundary Method Discretisations of the Stokes Problem

The Shifted Boundary Method (SBM) sidesteps body-fitted meshing by shifting boundary conditions onto a surrogate boundary and correcting for the displacement through Taylor expansions. Despite its broad analysis and application, scalable iterative solvers for the incompressible Stokes equations remain underdeveloped. We present a block preconditioner for SBM--Stokes discretisations that uses the velocity block together with a pressure mass matrix as a Schur complement approximation. Because the SBM system is non-symmetric, classical operator preconditioning does not apply directly; a field-of-values analysis instead shows that the non-symmetric SBM contributions act as asymptotically small perturbations of a standard saddle-point operator, yielding mesh-independent GMRES convergence on sufficiently fine meshes. Numerical experiments demonstrate iteration counts under refinement across geometries of increasing complexity. We expose a coarse-mesh regime in which an under-resolved grid produces elevated iteration counts, an artefact of insufficient resolution that vanishes once the mesh captures the geometry.