A short philosophical note on the origin of smoothed aggregations. (English) Zbl 1340.65295

Fraňková, Pavla; Hanuš, Milan; Kopincová, Hana; Kužel, Roman; Vaněk, Petr; Vastl, Zbyněk

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:

\(\bullet\)

Algorithm 1: The variational two-level algorithm with the post-smoothing step.

\(\bullet\)

Algorithm 2: The final error is minimized taking into account the pre-smoother.

\(\bullet\)

Algorithm 3: Minimization of the final error after performing the pre-smoothing, the coarse grid correction, and the post-smoothing.

\(\bullet\)

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].

Reviewer: Drahoslava Janovská (Praha)

MSC:

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

Keywords:

two-level method; pre-smoothing; coarse-grid correction; post-smoothing; error propagation; non-scalar elliptic problems; system of linear algebraic equations; symmetric positive definite matrix; smoothing; algorithms; aggregation method