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An algebraic multigrid method for elasticity based on an auxiliary topology with edge matrices
Author(s) -
Kogler Lukas,
Schöberl Joachim
Publication year - 2022
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.2408
Subject(s) - multigrid method , mathematics , finite element method , linear elasticity , partial differential equation , linear system , algebraic equation , algebraic number , scalability , elliptic partial differential equation , elasticity (physics) , vertex (graph theory) , robustness (evolution) , mathematical optimization , topology (electrical circuits) , computer science , nonlinear system , mathematical analysis , discrete mathematics , graph , biochemistry , chemistry , physics , materials science , quantum mechanics , database , combinatorics , gene , composite material , thermodynamics
This article introduces a novel approach to algebraic multigrid methods for large systems of linear equations coming from finite element discretizations of certain elliptic second‐order partial differential equations. Based on a discrete energy made up of edge and vertex contributions, we are able to develop coarsening criteria that guarantee two‐level convergence even for systems of equations such as linear elasticity . This energy also allows us to construct prolongations with prescribed sparsity pattern that still preserve kernel vectors exactly. These allow for a straightforward optimization that simplifies parallelization and reduces communication on coarse levels. Numerical experiments demonstrate efficiency and robustness of the method and scalability of the implementation.