Modified ℋ︁‐matrices for convection dominated problems

Abstract

AbstractHierarchical matrices (ℋ︁‐matrices) provide a technique for the sparse approximation of large, fully populated matrices. These matrices are 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 L∞‐coefficients, 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 [1]. The standard partitioning and admissibility condition fail to provide satisfactory results when applied to the singularly perturbed convection‐diffusion equation. We will introduce a modified (hierarchical) partitioning of the index and block index sets together with a modified admissibility condition, both depending on the (constant) convection direction. Numerical results will illustrate the positive effect of the changes.