Methods and Algorithms for Scientific Computing

Parallel Auxiliary Space AMG Solver for $H(div)$ Problems


In this paper we present a family of scalable preconditioners for matrices arising in the discretization of $H(div)$ problems using the lowest order Raviart--Thomas finite elements. Our approach belongs to the class of “auxiliary space''--based methods and requires only the finite element stiffness matrix plus some minimal additional discretization information about the topology and orientation of mesh entities. We provide a detailed algebraic description of the theory, parallel implementation, and different variants of this parallel auxiliary space divergence solver (ADS) and discuss its relations to the Hiptmair--Xu (HX) auxiliary space decomposition of $H(div)$ [SIAM J. Numer. Anal., 45 (2007), pp. 2483--2509] and to the auxiliary space Maxwell solver AMS [J. Comput. Math., 27 (2009), pp. 604--623]. An extensive set of numerical experiments demonstrates the robustness and scalability of our implementation on large-scale $H(div)$ problems with large jumps in the material coefficients.


  1. parallel algebraic multigrid
  2. $H(div)$ problems
  3. Raviart--Thomas elements
  4. auxiliary space preconditioning

MSC codes

  1. 65F10
  2. 65N30
  3. 65N55

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: A3079 - A3098
ISSN (online): 1095-7197


Submitted: 19 December 2011
Accepted: 15 October 2012
Published online: 18 December 2012


  1. parallel algebraic multigrid
  2. $H(div)$ problems
  3. Raviart--Thomas elements
  4. auxiliary space preconditioning

MSC codes

  1. 65F10
  2. 65N30
  3. 65N55



Panayot S. Vassilevski

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