Examination and Significance of Sparse Preconditioners for High-Order Finite Element Systems

Hundreds of millions of dollars have been committed toward the study of complex natural phenomena on today¿s massively parallel computers. Access to such computing power is enabling scientists to employ highly-accurate high-order finite element methods to solve previously intractable problems. However, these high-order finite element methods present new challenges to existing solution methods, because of fundamental differences in corresponding matrices and the need for higher memory consumption. This project will investigate the use of algebraic multi-grid preconditioners generated from sparser matrices as a cheaper alternative to the algebraic multi-grid preconditioners generated from high-order finite element matrices. Phase I compared the two algebraic multi-grid approaches: (1) the original high-order finite element matrix, and (2) a sparser matrix equivalent to using tri-linear finite elements on a mesh of equivalent order. It was demonstrated that the sparser approaches yield faster simulation times and reduced memory costs for most problems of interest. In Phase II, new capabilities will be added to two codes (HYPRE and PETSc), enabling users to construct a sparse approximation of a dense matrix generated from high-order finite element discretizations. This matrix will be used to construct cheaper algebraic multigrid-based preconditioners that still will enable fast simu­lations with nearly optimal convergence behavior. Commercial Applications and other Benefits as described by the awardee: DOE projects employing high-order finite elements should gain greater efficiency in their simulations on today¿s supercomputers when using these preconditioners. In addition, the new computational approach should generate consulting opportunities to assist users in optimally employing these preconditioners in their code