A parallel geometric multigrid method for finite elements on octree meshes
- ORNL
- University of Texas, Austin
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 {Omicron}(N/n{sub p} log N/n{sub p}) + {Omicron}(n{sub p} log n{sub p}), where N is the number of octree nodes and n{sub 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.
- Research Organization:
- Oak Ridge National Laboratory (ORNL); Center for Computational Sciences
- Sponsoring Organization:
- NE USDOE - Office of Nuclear Energy
- DOE Contract Number:
- AC05-00OR22725
- OSTI ID:
- 1033541
- Journal Information:
- SIAM Journal on Scientific Computing, Journal Name: SIAM Journal on Scientific Computing Journal Issue: 3 Vol. 32; ISSN 1064-8275; ISSN SJOCE3
- Country of Publication:
- United States
- Language:
- English
Similar Records
Persistent Octrees for Parallel Mesh Refinement Through Non-Volatile Byte-Addressable Memory
Low-Cost Parallel Algorithms for 2:1 Octree Balance