On the sparsity patterns of hierarchical finite element matrices

Abstract

The performance of preconditioned conjugate gradient methods for the solution of a linear system of equations H x=b depends strongly on the quality of the preconditioner. In many applications the system of equations to be solved originates from a partial differential equation discretized by the choice of an initial mesh geometry, a meshrefinement technique and a type of finite element basisfunctions. In general the resulting matrix is a sparse matrix which sparsity pattern only depends on the discretization choices. For the construction of a preconditioner only the matrix entries are needed, but investigations so far clearly have shown that taking into account additionally the discretization choices via the sparsity pattern leads to more effective preconditioning techniques, of which many exist.

Relatively simple techniques like ILU and SSOR take the matrix entries into account but disregard the sparsity pattern. Somewhat more effective techniques like Block-Incomplete ones make use of the sparsity patterns regularity resulting from a regular mesh geometry. In addition multi-grid techniques take into account that a fine mesh geometry is obtained by the refinement of a coarse mesh. Intermediate ‘levels’ of refinement are distinguished and used explicitly. More flexible are algebraic multilevel preconditioners which assign a ‘level’ to each individual degree of freedom.

As the sparsity pattern is of importance for the construction of good preconditioners it is analysed for the hierarchical matrix H resulting from a given discretization. The hierarchy is induced by the mesh refinement method applied. It is shown that the sparsity pattern is irregular but well structured in general and a simple refinement method is presented which enables a compact storage and quick retrieval of the matrix entries in the computers memory. An upperbound for the C.-B.-S. scalar for this method is determined to demonstrate that it is well suited for multilevel preconditioning and it is shown to have satisfying angle bounds. Further it turns out that the hierarchical matrix may be partially constructed in parallel, is block structured and shows block decay rates.

research supported by the Netherlands Organization for Scientific Research N.W.O.