ℋ︁‐matrices for the convection‐diffusion equation

Abstract

AbstractHierarchical matrices (ℋ︁‐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 applications where the kernel function displays certain smoothness properties. The error estimates for an approximation of the kernel function by a separable function can be carried over directly to error estimates for an approximation of the stiffness matrix by an ℋ︁‐matrix, using a certain standard partitioning and admissibility condition for matrix blocks. Similarly, ℋ︁‐matrix techniques can be applied in the finite element context where it is the inverse of the stiffness matrix that is fully populated. Here one needs a separable approximation of Green's function of the underlying boundary value problem in order to prove approximability by matrix blocks of low rank. Unfortunately, Green's function for the convection‐diffusion equation does not satisfy the required smoothness properties, hence prohibiting a straightforward generalization of the separable approximation through Taylor polynomials. We will use Green's function to motivate a modification in the (hierarchical) partitioning of the index set and as a consequence the resulting hierarchy of block partitionings as well as the admissibility condition. We will illustrate the effect of the proposed modifications by numerical results.