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A short philosophical note on the origin of smoothed aggregations. (English) Zbl 1340.65295
Brandts, J. (ed.) et al., Proceedings of the international conference ‘Applications of mathematics’, Prague, Czech Republic, May 15–17, 2013. In honor of the 70th birthday of Karel Segeth. Prague: Academy of Sciences of the Czech Republic, Institute of Mathematics (ISBN 978-80-85823-61-5). 67-76 (2013).
The authors solve a system of linear algebraic equations with a symmetric positive definite matrix \(A\) by a two-level method. In general, the multilevel method consists in combination of a coarse-grid correction and smoothing. In fact, the standard two-level method minimizes the error in an intermediate stage of the iteration. The authors suggest to minimize the error after coarse-grid correction and subsequent smoothing. This idea leads to a minimization problem for the error of the approximation. The authors extend the result to the two-level method with the pre-smoothing, the coarse grid correction, and the post-smoothing.
Four algorithms are given:
Algorithm 1: The variational two-level algorithm with the post-smoothing step.
Algorithm 2: The final error is minimized taking into account the pre-smoother.
Algorithm 3: Minimization of the final error after performing the pre-smoothing, the coarse grid correction, and the post-smoothing.
Algorithm 4: The generalized aggregation method suitable for non-scalar elliptic problems.

The paper ends with the test 3D anisotropic problem.
For the entire collection see [Zbl 1277.00032].
65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N15 Error bounds for boundary value problems involving PDEs
35J25 Boundary value problems for second-order elliptic equations
65F10 Iterative numerical methods for linear systems
65D10 Numerical smoothing, curve fitting
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