Efficient Parallel Adaptive Finite Element Methods Using Self-Scheduling Data and Computations
HiPC '99 Proceedings of the 6th International Conference on High Performance Computing
FOCS '99 Proceedings of the 40th Annual Symposium on Foundations of Computer Science
A Cache-Aware Algorithm for PDEs on Hierarchical Data Structures Based on Space-Filling Curves
SIAM Journal on Scientific Computing
A Parallel Geometric Multigrid Method for Finite Elements on Octree Meshes
SIAM Journal on Scientific Computing
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The most efficient numerical methods for the solution of partial differential equations, multigrid methods on adaptively refined grids, imply several drawbacks from the point of view of memory-efficiency on high-performance computer architectures: First, we loose the trivial structure expressed by the simple i,j-indexing of grid points or cells. This effect is even worsened by the usage of hierarchical data and – if implemented in a naive way – leads to both increased storage requirements (neighbourhoodrelations possibly modified difference stencils) and a less efficient data access (worse locality of data and additional data dependencies), in addition. Our approach to overcome this quandary between numerical and hardware-efficiency relies on structured but still highly flexible adaptive grids, the so-called space-partitioning grids, cell-oriented operator evaluations, and the construction of very efficient data structures based on the concept of space-filling curves. The focus of this paper is in particular on the technical and algorithmical details concerning the interplay between data structures, space-partitioning grids and space-filling curves.