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

Authors:
Maxime Theillard;Chris H. Rycroft;Frédéric Gibou
Affiliations:
Department of Mechanical Engineering, University of California, Santa Barbara, USA 93106;Department of Mathematics, University of California and LBNL, Berkeley, USA 94720 and Department of Mathematics, Lawrence Berkeley Laboratory, Berkeley, USA 94720;Department of Mechanical Engineering, University of California, Santa Barbara, USA 93106 and Department of Computer Science, University of California, Santa Barbara, USA 93106
Venue:
Journal of Scientific Computing
Year:
2013

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Abstract

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.