Coupled clustering strategies for hierarchical matrix preconditioners in saddle point problems

Fluid flow problems can be modeled by the Navier–Stokes or, after linearization, by the Oseen equations. Their discretization results in discrete saddle point problems. These systems of equations are typically very large and need to be solved iteratively. Standard (block-) preconditioning techniques for saddle point problems rely on an approximation of the Schur complement. Such an approximation can be obtained by a hierarchical (H-) matrix LU-decomposition, which first approximates the Schur complement explicitly. The computational complexity of this computation depends, among other things, on the hierarchical block structure of the involved matrices. However, widely used techniques do not consider the connection between the discretization grids for the velocity field and the pressure, respectively. Here, we present a hierarchical block structure for the finite element discretization of the gradient operator that is improved by considering the connection between the two involved grids. Numerical results imply that the improved block structure allows for a faster computation of the Schur complement, which is the bottleneck for the set-up of the H-matrix LU-decomposition.