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Improving the finite element ordering for the frontal solver. (English) Zbl 0979.65109

The context is a domain decomposition method and the linear systems in each subdomain have medium size. The optimization of the numbering of nodes in each subdomain can have different goals:
1. When a direct method (Cholesky, Gauss) is chosen along with skyline structure of the matrix, one has to minimize the bandwidth of the overall system.
2. A direct method with sparse data structure needs to maximize the number of zero terms after triangulation. The sequence of nodes optimal for this pattern is usually different from that obtained for the skyline (band) storage.
3. When a frontal solver is used, the optimization concerning the sequence of elements is sought in the way to minimize the frontwidth.
The paper is concerned with the proposition of two improvements of greedy methods. A wave reordering method adapts the reordering strategy during the process, and the tabu search optimization technique is adapted for finite element reordering. The well-known benchmark problems from 1979 are considered as small for actual purposes, and so numerical tests are not restricted to them.

MSC:

65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F05 Direct numerical methods for linear systems and matrix inversion
65F50 Computational methods for sparse matrices
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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