H-matrix Preconditioners in Convection-Dominated Problems

Hierarchical matrices provide a data-sparse way to approximate fully populated matrices. In this paper we exploit H-matrix techniques to approximate the LU -decompositions of stiffness matrices as they appear in (finite element or finite difference) discretizations of convectiondominated elliptic partial differential equations. These sparse H-matrix approximations may then be used as preconditioners in iterative methods. Whereas the approximation of the matrix inverse by an H-matrix requires some modification in the underlying index clustering when applied to convectiondominant problems [11], the H-LU-decomposition works well in the standard H-matrix setting even in the convection dominant case. We will complement our theoretical analysis with some numerical examples.