Structuring preconditioners for unstructured meshes

An application of a fictitious space technique to the construction of preconditioners for stiffness matrices generated by finite element method on completely unstructured meshes is considered in the paper. Numerical tests for model elliptic operators are presented. In recent years a number of efficient multilevel techniques [1,3,4,11] have been proposed for either solving or preconditioning systems of grid equations approximating second-order elliptic boundary value problems. Many of these methods are widely used in research and industrial problems. However, multilevel and multigrid methods deal with meshes possessing certain structure or hierarchy. It is the hierarchy which provides efficiency of the method. The multilevel methods, in general, reduce the problem to be solved on a complicated or rather fine mesh to a set of problems which are easy to solve. In many scientific and engineering applications there is a necessity to solve problems in domains with complicated geometry where it is very hard or even impossible to construct a hierarchical grid. Thus, the finite element discretizations should be performed on completely unstructured grids composed either of triangles in 2D or of tetrahedra in 3D. This is also dictated by available adaptive mesh generators used for discretization of computational domain, which allows a significant reduction of the number of grid nodes (i.e. number of unknowns) while maintaining the quality of approximation. Recently, several efficient multigrid-type algorithms for finite element elliptic systems associated with the unstructured grid have been presented in [2,5]. They are based on the specific construction of the grid hierarchy for a given unstructured mesh and the application of a multigrid scheme on the produced sequence of grids. For preconditioning a discrete model operator associated with unstructured triangulation we apply several types of multilevel structured preconditioners taking advantage of the fictitious space technique [18]. Given an unstructured mesh, we generate a structured hierarchical grid which 'approximates' the original mesh in some sense. With special interpolation operators we reduce the problem of constructing a preconditioner in a space associated with the unstructured mesh to the problem of constructing the preconditioner in a fictitious space corresponding to the structured grid. Within a fictitious space we apply multilevel preconditioners of three types: domain decomposition, multigrid, and BPX. A multilevel local refinement preconditioner [3,21] has been chosen as one of domain decomposition algorithms. An algebraic multigrid/substructuring preconditioner [8,11] and BPX preconditioner [4] * The work was supported hi part by the Dassault Aviation and the Russian Foundation for the Basic Research (94-01-01204). ' Institute of Numerical Mathematics, Russian Academy of Sciences, Moscow 117334, Russia 140 KG. Dyadechko, Yu . I. Wash , and Yu . V. Vassilevski present a multigrid technique and an additive multilevel method, respectively. Since each of the above preconditioners is defined for its own type of structured mesh and fictitious space, we compare the efficiency of the resulting preconditioners on unstructured meshes in terms of the experimental condition number of the preconditioned stiffness matrix and the ratio of CPU time for solving a system with a preconditioner to CPU time for stiffness matrix-vector multiplication. The paper is organized as follows. In Section 1 we pose the problem to be considered and introduce a fictitious space technique. In Sections 2, 3, and 4 we give descriptions of multilevel methods and related fictitious spaces for the case of algebraic multigrid, BPX, and multilevel local refinement preconditioners, respectively. In Section 5 when considering several types of completely unstructured regular triangulations of a unit square, we make a comparison of the results of numerical experiments for the above preconditioners. 1. FICTITIOUS SPACE TECHNIQUE Let Π be a unit square and U be its regular unstructured triangulation [6]. Define a finite element space V of functions which are continuous in 77, linear at each triangle T. from Π and vanish at 377. Let NQ be a number of interior nodes in 77^ and let Λ^χΛ^0 symmetric positive definite stiffness matrix AQ be defined by