Iterative Stokes Solvers in the Harmonic Velte Subspace

Abstract

We explore the prospects of utilizing the decomposition of the function space (H ¹ ₀)ⁿ (where n=2,3) into three orthogonal subspaces (as introduced by Velte) for the iterative solution of the Stokes problem. It is shown that Uzawa and Arrow-Hurwitz iterations – after at most two initial steps – can proceed fully in the third, smallest subspace. For both methods, we also compute optimal iteration parameters. Here, for two-dimensional problems, the lower estimate of the inf-sup constant by Horgan and Payne proves useful and provides an inclusion of the spectrum of the Schur complement operator of the Stokes problem.

We further consider the conjugate gradient method in the third Velte subspace and derive a corresponding convergence estimate.

Computational results show the effectiveness of this approach for discretizations which admit a discrete Velte decomposition.