Coupled clustering in hierarchical matrices for the oseen problem

Fluid flow problems can be modelled 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 ((Formula presented.) -) 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 hierarchical 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 which is improved by considering the connection between the two involved grids. We prove a logarithmic depth estimate for cluster trees generated with the coupled clustering. Numerical results will show that the improved block structure allows for a faster computation of the Schur complement which is the bottleneck for the set-up of the (Formula presented.) -matrix LU-decomposition. The presented coupled clustering is also applicable to other types of mixed (finite element) problems.

Subjects

clustering

hierarchical matrix

Oseen problem

preconditioning

DDC Class

530.42: Fluid Physics

518: Numerical Analysis

Funding(s)

Projekt DEAL

Lizenz

https://creativecommons.org/licenses/by/4.0/

Publication version

publishedVersion

Name

Numerical Methods in Fluids - 2026 - Grams - Coupled Clustering in Hierarchical Matrices for the Oseen Problem.pdf

Type

Main Article

Size

1.37 MB

Format

Adobe PDF

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Coupled clustering in hierarchical matrices for the oseen problem