FRIDAY, April 18, 2003
Time: 2:00 - 3:00 PM
Constant Hall 1052

Dr. Sabine Leborne
Tennessee Technological University

Hierarchical matrices (H-matrices) provide a technique for the sparse approximation of large, fully populated matrices. This technique has been shown to be applicable to stiffness matrices arising in boundary element method (BEM) and finite element method (FEM) applications. In the latter case, it is the inverse stiffness matrix that is fully populated and approximated by an H-matrix.

The H-matrix approximation is based on an approximation of the kernel function (BEM) or Green's function (FEM) by a separable function. A typical H-matrix approximant is constructed by using an algorithm that iteratively partitions the (block) index set until a certain admissibility condition is satisfied for an index block. For elliptic operators with coefficients in the space L(infinity), it has been shown that the standard partitioning algorithm in connection with the standard admissibility condition lead to hierachical matrices that approximate the (inverse) of the stiffness matrix with an error of the same order as the discretization order while having nearly optimal storage complexity.

In this talk, we will introduce the construction of H-matrices in the standard setting together with some complexity and approximation properties. We will then demonstrate the shortcomings of applying the standard partitioning and admissibility condition to the singularly perturbed convection-diffusion equation. We will construct a modified (hierarchical) partitioning of the index and block index sets together with a modified admissibility condition, both depending on the (constant) convection direction and the grid parameter h. An important observation is that the H-matrix approximant benefits from the convection dierction aligning with the underlying grid. Numerical results will illustrate the effect of these changes.