MiS Preprint Repository

We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV ( that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

MiS Preprint

${\cal H}$-matrix preconditioners in convection-dominated problems

Sabine Le Borne and Lars Grasedyck


Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. In this paper we exploit ${\cal H}$-matrix techniques to approximate the $LU$-decompositions of stiffness matrices as they appear in (finite element or finite difference) discretizations of convection-dominated elliptic partial differential equations.

These sparse ${\cal H}$-matrix approximations may then be used as preconditioners in iterative methods. Whereas the approximation of the matrix inverse by an ${\cal H}$-matrix requires some modification in the underlying index clustering when applied to convection-dominant problems, the ${\cal H}$-LU-decomposition works well in the standard ${\cal H}$-matrix setting even in the convection dominant case. We will complement our theoretical analysis with some numerical examples.

MSC Codes:
65F05, 65F30, 65F50
hierarchical matrices, preconditioning, convection-dominant problems

Related publications

2006 Repository Open Access
Sabine LeBorne and Lars Grasedyck

\(\mathscr {H}\)-matrix preconditioners in convection-dominated problems

In: SIAM journal on matrix analysis and applications, 27 (2006) 4, pp. 1172-1183