Talk on Chemnitz Finite Element Symposium 2018: On a commutator preconditioner for the Navier-Stokes equations

We provide a novel analysis of the PCD (pressure convection-diffusion) preconditioner based on infinite-dimensional interpretation of the underlying operators, which is the basis of operator preconditioning theory. Our results help to answer open questions concerning practical implementation and performance. In particular, we clarify the choice of boundary conditions which is, to our knowledge, a problem missing rigorous analysis, although some heuristic treatment has appeared; see [1, 2]. Moreover, our approach provides a unified theory for both variants (order of the commuted operators) which have thus far only been treated separately in the literature. Last but not least, our infinite-dimensional analysis reveals Fredholm structure in the preconditioner which must be respected in the discretization; failure to do so results in significant deterioration of performance, which has previously been observed but not correctly identified and resolved.[1] Howard C. Elman, David J. Silvester, and Andrew J. Wathen. Finite elements and fast iterative solvers: with applications in incompressible fluid dynamics. 2nd. Oxford University Press, 2014, Section 9.3.2[2] Maxim A. Olshanskii and Yuri V. Vassilevski. “Pressure Schur Complement Preconditioners for the Discrete Oseen Problem”. In: SIAM Journal on Scientific Computing 29.6 (2007), pp. 2686–2704. DOI: 10.1137/070679776, Section 2.1