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Incremental unknowns in finite differences: Condition number of the matrix. (English) Zbl 0773.65080
The authors apply the method of incremental unknowns to multilevel finite difference approximations of linear second order elliptic boundary value problems. Roughly speaking, in a two-level setting the incremental unknowns consist of the nodal values at the coarse grid points and, for the fine grid points, of the increment to the averaged value a the neighbouring coarse grid points.
Evidently, there is a close relationship to H. Yserentant’s hierarchical basis finite element method [Numer. Math. 49, 379-412 (1986; Zbl 0608.65065)] but it should be emphasized that the two methods are not the same. In case of the Dirichlet problem and spatial step size \(h\) it is shown that the condition number behaves like \(O((\log h)^ 2)\) compared to \(O(h^{-2})\) for the standard nodal unknowns. The theoretical results are supported by several numerical examples.

MSC:
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65F35 Numerical computation of matrix norms, conditioning, scaling
65N06 Finite difference methods for boundary value problems involving PDEs
65F10 Iterative numerical methods for linear systems
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
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