Journal of Scientific Computing

, Volume 55, Issue 1, pp 1–15 | Cite as

A Multigrid Method on Non-Graded Adaptive Octree and Quadtree Cartesian Grids

  • Maxime Theillard
  • Chris H. Rycroft
  • Frédéric Gibou


In order to develop efficient numerical methods for solving elliptic and parabolic problems where Dirichlet boundary conditions are imposed on irregular domains, Chen et al. (J. Sci. Comput. 31(1):19–60, 2007) presented a methodology that produces second-order accurate solutions with second-order gradients on non-graded quadtree and octree data structures. These data structures significantly reduce the number of computational nodes while still allowing for the resolution of small length scales. In this paper, we present a multigrid solver for this framework and present numerical results in two and three spatial dimensions that demonstrate that the computational time scales linearly with the number of nodes, producing a very efficient solver for elliptic and parabolic problems with multiple length scales.


Multigrid method Poisson’s equation Non-graded adaptive grid Octrees Quadtrees Second-order discretization Complex geometry 



M. Theillard and F. Gibou were supported by the Department of Energy under contract number DE-FG02-08ER15991; by the Office of Naval Research through contract N00014-11-1-0027; the National Science Foundation through contract CHE-1027817; the W.M. Keck Foundation; and by the Institute for Collaborative Biotechnologies through contract W911NF-09-D-0001 from the U.S. Army Research Office. C.H. Rycroft was supported by the Director, Office of Science, Computational and Technology Research, U.S. Department of Energy under contract number DE-AC02-05CH11231.


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • Maxime Theillard
    • 1
  • Chris H. Rycroft
    • 2
    • 3
  • Frédéric Gibou
    • 1
    • 4
  1. 1.Department of Mechanical EngineeringUniversity of CaliforniaSanta BarbaraUSA
  2. 2.Department of MathematicsUniversity of California and LBNLBerkeleyUSA
  3. 3.Department of MathematicsLawrence Berkeley LaboratoryBerkeleyUSA
  4. 4.Department of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA

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