# Low-rank correction of the Schur complement inverses for the Oseen problem

Motivation: Flow problems are modelled by the Navier-Stokes equations, and their numerical solution (discretisation and non-linear solver) leads to the so-called Oseen problem. This is a system of equations with a large, weakly populated, indefinite and non-symmetric matrix of the type [A B; B’ 0]. Many solution methods are based on a matrix decomposition, which must solve a so-called Schur complement system Sx=b with the Schur complement matrix S=B’ A^{-1} B as a subproblem.
The aim is to investigate whether approximations of the Schur complement inverses S^{-1}_{app} (which are relatively easy to calculate or apply) can be efficiently improved by low-rank corrections.

Methods: Various approaches exist in the literature for constructing approximations of the Schur complement or its inverse. Let S_i be such an approximation of the inverse. Then the hope is that S*S_i is close to the unit matrix I in the sense that I-S*S_i is of small rank or has rapidly decreasing singular values. So there exist rectangular matrices U,V, which provide a low rank approximation of I-S*S_i by UV’. From UV’ approx I-S*S_i one concludes S*S_i approx I-UV’ or S^{-1} approx S_i (I-UV’)^{-1}. The latter inverse can be efficiently transformed via the Shermann-Morrison-Woodbury formula to S^{-1} approx S_i (I+XY’) with low rank matrices X, Y.
In this project, on the one hand, it is to be investigated for which approximations S_i the hope that I-S*S_i has rapidly decreasing singular values is confirmed. On the other hand, different numerical methods are to be developed.
to determine the resulting rectangular matrices U, V. Randomised factorisations, adaptive cross approximation (ACA) and Arnoldi methods offer approaches to this. The approaches are to be applied to various test problems and approximations of the Schur complement inverses in extensive numerical tests and thus compared.

Project goals and work packages:
– Literature research on the Oseen problem and Schur complement approximations
– Familiarisation with programme packages for generating model problems
– Implementation (C/C++) of numerical methods for the determination of low rank approximations U, V given the approximation S_i – Use of the new preconditioner S_i (I+XY’) for the iterative solution of the Schur complement problem

– Extensive testing and documentation of numerical results in terms of convergence rates, computational and memory requirements;