Proceedings of the 2010 ACM/IEEE International Conference for High Performance Computing, Networking, Storage and Analysis
p4est: Scalable Algorithms for Parallel Adaptive Mesh Refinement on Forests of Octrees
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
A highly scalable matrix-free multigrid solver for μFE analysis based on a pointer-less octree
LSSC'11 Proceedings of the 8th international conference on Large-Scale Scientific Computing
Parallel geometric-algebraic multigrid on unstructured forests of octrees
SC '12 Proceedings of the International Conference on High Performance Computing, Networking, Storage and Analysis
Journal of Scientific Computing
A Multigrid Method on Non-Graded Adaptive Octree and Quadtree Cartesian Grids
Journal of Scientific Computing
Cluster optimization and parallelization of simulations with dynamically adaptive grids
Euro-Par'13 Proceedings of the 19th international conference on Parallel Processing
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In this article, we present a parallel geometric multigrid algorithm for solving variable-coefficient elliptic partial differential equations on the unit box (with Dirichlet or Neumann boundary conditions) using highly nonuniform, octree-based, conforming finite element discretizations. Our octrees are 2:1 balanced, that is, we allow no more than one octree-level difference between octants that share a face, edge, or vertex. We describe a parallel algorithm whose input is an arbitrary 2:1 balanced fine-grid octree and whose output is a set of coarser 2:1 balanced octrees that are used in the multigrid scheme. Also, we derive matrix-free schemes for the discretized finite element operators and the intergrid transfer operations. The overall scheme is second-order accurate for sufficiently smooth right-hand sides and material properties; its complexity for nearly uniform trees is $\mathcal{O}(\frac{N}{n_p}\log\frac{N}{n_p})+\mathcal{O}(n_p\log n_p)$, where $N$ is the number of octree nodes and $n_p$ is the number of processors. Our implementation uses the Message Passing Interface standard. We present numerical experiments for the Laplace and Navier (linear elasticity) operators that demonstrate the scalability of our method. Our largest run was a highly nonuniform, 8-billion-unknown, elasticity calculation using 32,000 processors on the Teragrid system, “Ranger,” at the Texas Advanced Computing Center. Our implementation is publically available in the Dendro library, which is built on top of the PETSc library from Argonne National Laboratory.