We address the numerical solution of linear systems arising from the hybrid discretizations of second-order elliptic partial differential equations. Such discretizations hinge on a hybrid set of degrees of freedom (DoFs), respectively, defined in cells and faces, which naturally gives rise to a global hybrid system of linear equations. Assuming that the cell unknowns are only locally coupled, they can be efficiently eliminated from the system, leaving only face unknowns in the resulting Schur complement, which is also called the statically condensed matrix. We propose in this work an algebraic multigrid (AMG) preconditioner specifically targeting condensed systems corresponding to lowest-order discretizations (piecewise constant). Like traditional AMG methods, we retrieve geometric information on the coupling of the DoFs from algebraic data. However, as the condensed matrix only gives information on the faces, we use the uncondensed version to reconstruct the connectivity graph between elements and faces. An aggregation-based coarsening strategy mimicking a geometric coarsening or semicoarsening can then be set up to build coarse levels. Numerical experiments are performed on diffusion problems discretized by the hybrid high-order method at the lowest order. Our approach uses a K-cycle to precondition an outer flexible Krylov method. The results demonstrate similar performances, in most cases, compared to a standard AMG method and a notable improvement on anisotropic problems with Cartesian meshes.


  1. algebraic multigrid
  2. hybrid methods
  3. static condensation

MSC codes

  1. 65N55
  2. 65N22
  3. 65F50
  4. 65F08

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Information & Authors


Published In

cover image SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Pages: S329 - S350
ISSN (online): 1095-7197


Submitted: 28 June 2021
Accepted: 8 September 2022
Published ahead of print: 3 March 2023
Published online: 26 June 2023


  1. algebraic multigrid
  2. hybrid methods
  3. static condensation

MSC codes

  1. 65N55
  2. 65N22
  3. 65F50
  4. 65F08



IMAG, University of Montpellier, CNRS, Montpellier, France.
EDF R&D, Paris-Saclay, France.
IMAG, University of Montpellier, CNRS, Montpellier, France; IRIT, Toulouse, France; CERFACS, Toulouse, France; and FAU, Erlangen-Nürnberg, Germany.
Ulrich Rüde
CERFACS, Toulouse, France, and FAU, Erlangen-Nürnberg, Germany.

Funding Information

Funding: This work was supported by ANR project Fast4HHO under contract ANR-17-CE23-0019.

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