Finite-element preconditioning for pseudospectral solutions of elliptic problems
SIAM Journal on Scientific and Statistical Computing
How fast are nonsymmetric matrix iterations
SIAM Journal on Matrix Analysis and Applications
Choosing the forcing terms in an inexact Newton method
SIAM Journal on Scientific Computing - Special issue on iterative methods in numerical linear algebra; selected papers from the Colorado conference
Enhanced Nonlinear Iterative Techniques Applied to a Nonequilibrium Plasma Flow
SIAM Journal on Scientific Computing
An adaptive refinement strategy for hp-finite element computations
Proceedings of international centre for mathematical sciences on Grid adaptation in computational PDES : theory and applications: theory and applications
A Note on Preconditioning for Indefinite Linear Systems
SIAM Journal on Scientific Computing
Performance modeling and tuning of an unstructured mesh CFD application
Proceedings of the 2000 ACM/IEEE conference on Supercomputing
BoomerAMG: a parallel algebraic multigrid solver and preconditioner
Applied Numerical Mathematics - Developments and trends in iterative methods for large systems of equations—in memoriam Rüdiger Weiss
A Fully Automatic hp-Adaptivity
Journal of Scientific Computing
Jacobian-free Newton-Krylov methods: a survey of approaches and applications
Journal of Computational Physics
Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method
Journal of Scientific Computing
Algebraic multigrid for higher-order finite elements
Journal of Computational Physics
libMesh: a C++ library for parallel adaptive mesh refinement/coarsening simulations
Engineering with Computers
Journal of Computational Physics
Computing the First Eigenpair of the p-Laplacian via Inverse Iteration of Sublinear Supersolutions
Journal of Scientific Computing
Multiphysics simulations: Challenges and opportunities
International Journal of High Performance Computing Applications
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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.