Mathematics of Computation
Elliptic grid generation based on Laplace equations and algebraic transformations
Journal of Computational Physics
Mathematics of Computation
Implicit Extrapolation Methods for Variable Coefficient Problems
SIAM Journal on Scientific Computing
Parallel multigrid in an adaptive PDE solver based on hashing and space-filling curves
Parallel Computing - Special issue on parallelization techniques for numerical modelling
Parallel multigrid solver for 3D unstructured finite element problems
SC '99 Proceedings of the 1999 ACM/IEEE conference on Supercomputing
Achieving high sustained performance in an unstructured mesh CFD application
SC '99 Proceedings of the 1999 ACM/IEEE conference on Supercomputing
Multigrid for Locally Refined Meshes
SIAM Journal on Scientific Computing
A multigrid tutorial (2nd ed.)
A multigrid tutorial (2nd ed.)
Computer simulations of cardiac electrophysiology
Proceedings of the 2000 ACM/IEEE conference on Supercomputing
Performance modeling and tuning of an unstructured mesh CFD application
Proceedings of the 2000 ACM/IEEE conference on Supercomputing
Multigrid
Proceedings of the 1994 ACM/IEEE conference on Supercomputing
SuperLU_DIST: A scalable distributed-memory sparse direct solver for unsymmetric linear systems
ACM Transactions on Mathematical Software (TOMS)
Gerris: a tree-based adaptive solver for the incompressible Euler equations in complex geometries
Journal of Computational Physics
Proceedings of the 2004 ACM/IEEE conference on Supercomputing
High Resolution Forward And Inverse Earthquake Modeling on Terascale Computers
Proceedings of the 2003 ACM/IEEE conference on Supercomputing
Improving the computational intensity of unstructured mesh applications
Proceedings of the 19th annual international conference on Supercomputing
Scalable Parallel Octree Meshing for TeraScale Applications
SC '05 Proceedings of the 2005 ACM/IEEE conference on Supercomputing
Is 1.7 x 10^10 Unknowns the Largest Finite Element System that Can Be Solved Today?
SC '05 Proceedings of the 2005 ACM/IEEE conference on Supercomputing
High Resolution Aerospace Applications using the NASA Columbia Supercomputer
SC '05 Proceedings of the 2005 ACM/IEEE conference on Supercomputing
An Introduction to Algebraic Multigrid
Computing in Science and Engineering
Shape Representation and Classification Using the Poisson Equation
IEEE Transactions on Pattern Analysis and Machine Intelligence
From mesh generation to scientific visualization: an end-to-end approach to parallel supercomputing
Proceedings of the 2006 ACM/IEEE conference on Supercomputing
Parallel-adaptive simulation with the multigrid-based software framework UG
Engineering with Computers
Journal of Computational Physics
A general fictitious domain method with immersed jumps and multilevel nested structured meshes
Journal of Computational Physics
Low-constant parallel algorithms for finite element simulations using linear octrees
Proceedings of the 2007 ACM/IEEE conference on Supercomputing
Bottom-Up Construction and 2:1 Balance Refinement of Linear Octrees in Parallel
SIAM Journal on Scientific Computing
Dendro: parallel algorithms for multigrid and AMR methods on 2:1 balanced octrees
Proceedings of the 2008 ACM/IEEE conference on Supercomputing
HPCC'06 Proceedings of the Second international conference on High Performance Computing and Communications
A parallel adaptive cartesian PDE solver using space–filling curves
Euro-Par'06 Proceedings of the 12th 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.