An algebraic multigrid method based on an auxiliary topology with edge matrices

Lukas Kogler, Joachim Schöberl

This paper 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. 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.

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