Journal of Computational Physics
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Timely Communication: Diagonal Edge Preconditioners in p-version and Spectral Element Methods
SIAM Journal on Scientific Computing
Preconditioning Chebyshev Spectral Collocation by Finite-Difference Operators
SIAM Journal on Numerical Analysis
A Fast and High Quality Multilevel Scheme for Partitioning Irregular Graphs
SIAM Journal on Scientific Computing
LAPACK Users' guide (third ed.)
LAPACK Users' guide (third ed.)
A finite element multigrid preconditioner for Chebyshev—collocation
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
Algorithm 583: LSQR: Sparse Linear Equations and Least Squares Problems
ACM Transactions on Mathematical Software (TOMS)
Fast parallel direct solvers for Coarse Grid problems
Journal of Parallel and Distributed Computing
Variational mesh adaptation: isotropy and equidistribution
Journal of Computational Physics
Algebraic Multigrid Based on Element Interpolation (AMGe)
SIAM Journal on Scientific Computing
Semi-Implicit Spectral Element Atmospheric Model
Journal of Scientific Computing
A Schwarz Preconditioner for the Cubed-Sphere
SIAM Journal on Scientific Computing
Hybrid Multigrid/Schwarz Algorithms for the Spectral Element Method
Journal of Scientific Computing
Algebraic multigrid for higher-order finite elements
Journal of Computational Physics
Hi-index | 31.45 |
High-order finite elements often have a higher accuracy per degree of freedom than the classical low-order finite elements. However, in the context of implicit time-stepping methods, high-order finite elements present challenges to the construction of efficient simulations due to the high cost of inverting the denser finite element matrix. There are many cases where simulations are limited by the memory required to store the matrix and/or the algorithmic components of the linear solver. We are particularly interested in preconditioned Krylov methods for linear systems generated by discretization of elliptic partial differential equations with high-order finite elements. Using a preconditioner like Algebraic Multigrid can be costly in terms of memory due to the need to store matrix information at the various levels. We present a novel method for defining a preconditioner for systems generated by high-order finite elements that is based on a much sparser system than the original high-order finite element system. We investigate the performance for non-uniform meshes on a cube and a cubed sphere mesh, showing that the sparser preconditioner is more efficient and uses significantly less memory. Finally, we explore new methods to construct the sparse preconditioner and examine their effectiveness for non-uniform meshes. We compare results to a direct use of Algebraic Multigrid as a preconditioner and to a two-level additive Schwarz method.