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

Today, higher-order polynomials are being employed more frequently in finite element discretizations, used in the simulation of previ¬ously intractable problems of importance to the DOE. The use of higher-order polynomials creates new challenges in efficiently solving the corresponding finite element matrix problems. Precondi¬tioners are used to achieve good convergence behavior; however, the design of preconditioners with minimal memory usage and minimal application cost is not trivial. Sparser precondition¬ers are needed that significantly reduce the overall memory consumption, but still maintain good convergence behavior. This project will develop preconditioners for high-order polynomials that have sparsity patterns equivalent to those generated by first-order polyno¬mials. These sparse preconditioners will be matrices of the same rank developed on a finite element mesh constructed from the Gauss nodes of the high-order polynomials. Commercial Applications and other Benefits as described by the awardee: The new preconditioners should enable DOE projects employing high-order ¿nite elements to achieve greater efficiency in their simulations on today¿s supercomputers. As users begin to employ these preconditioners in their codes, consulting opportunities would be expected to arise.