Efficient Nonlinear Solvers for Nodal High-Order Finite Elements in 3D
- 328 Downloads
Conventional high-order finite element methods are rarely used for industrial problems because the Jacobian rapidly loses sparsity as the order is increased, leading to unaffordable solve times and memory requirements. This effect typically limits order to at most quadratic, despite the favorable accuracy and stability properties offered by quadratic and higher order discretizations. We present a method in which the action of the Jacobian is applied matrix-free exploiting a tensor product basis on hexahedral elements, while much sparser matrices based on Q 1 sub-elements on the nodes of the high-order basis are assembled for preconditioning. With this “dual-order” scheme, storage is independent of spectral order and a natural taping scheme is available to update a full-accuracy matrix-free Jacobian during residual evaluation. Matrix-free Jacobian application circumvents the memory bandwidth bottleneck typical of sparse matrix operations, providing several times greater floating point performance and better use of multiple cores with shared memory bus. Computational results for the p-Laplacian and Stokes problem, using block preconditioners and AMG, demonstrate mesh-independent convergence rates and weak (bounded) dependence on order, even for highly deformed meshes and nonlinear systems with several orders of magnitude dynamic range in coefficients. For spectral orders around 5, the dual-order scheme requires half the memory and similar time to assembled quadratic (Q 2) elements, making it very affordable for general use.
KeywordsHigh-order Finite element method Newton-Krylov Preconditioning
Unable to display preview. Download preview PDF.
- 2.Balay, S., Buschelman, K., Eijkhout, V., Gropp, W.D., Kaushik, D., Knepley, M.G., Curfman McInnes, L., Smith, B.F., Zhang, H.: PETSc users manual. Technical Report ANL-95/11—Revision 3.0.0, Argonne National Laboratory (2008) Google Scholar
- 4.Blacker, T., Bohnhoff, W., Edwards, T., Hipp, J., Lober, R., Mitchell, S., Sjaardema, G., Tautges, T., Wilson, T., Oakes, W., et al.: CUBIT mesh generation environment. Technical report, Sandia National Labs., Albuquerque, NM. Cubit Development Team (1994) Google Scholar
- 11.Evans, L.C.: The 1-Laplacian, the ∞-Laplacian and differential games. Perspect. Nonlinear Partial Differ. Equ.: In Honor of Haim Brezis 446, 245 (2007) Google Scholar
- 12.Gee, M.W., Siefert, C.M., Hu, J.J., Tuminaro, R.S., Sala, M.G.: ML 5.0 smoothed aggregation user’s guide. Technical Report SAND2006-2649, Sandia National Laboratories (2006) Google Scholar
- 13.Gropp, W.D., Kaushik, D.K., Keyes, D.E., Smith, B.: Performance modeling and tuning of an unstructured mesh cfd application. In: Supercomputing ’00: Proceedings of the 2000 ACM/IEEE Conference on Supercomputing (CDROM), Washington, DC, USA, 2000, p. 34. IEEE Computer Society, New York (2000) Google Scholar
- 16.Interoperable technologies for advanced petascale simulations (ITAPS). http://www.itaps.org/
- 32.Tautges, T.J., Meyers, R., Merkley, K., Stimpson, C., Ernst, C.: MOAB: a mesh-oriented database. Technical report, Sandia National Laboratories, April 2004 Google Scholar