GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems
SIAM Journal on Scientific and Statistical Computing
Deflation of conjugate gradients with applications to boundary value problems
SIAM Journal on Numerical Analysis
Spectral element multigrid. I. Formulation and numerical results
Journal of Scientific Computing
Line relaxation for spectral multigrid methods
Journal of Computational Physics
Finite-element preconditioning for pseudospectral solutions of elliptic problems
SIAM Journal on Scientific and Statistical Computing
Domain decomposition preconditioning for p-version finite elements with high aspect rations
Applied Numerical Mathematics - II on Domain decomposition; Guest Editor: W. Proskurowski
Journal of Computational Physics
Preconditioning Legendre spectral collocation approximations to elliptic problems
SIAM Journal on Numerical Analysis
Spectral-element preconditioners for the Uzawa pressure operator applied to incompressible flows
Journal of Scientific Computing
SIAM Journal on Numerical Analysis
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Domain decomposition: parallel multilevel methods for elliptic partial differential equations
Journal of Computational Physics
Timely Communication: Diagonal Edge Preconditioners in p-version and Spectral Element Methods
SIAM Journal on Scientific Computing
Quasi-Optimal Schwarz Methods for the Conforming Spectral Element Discretization
SIAM Journal on Numerical Analysis
Overlapping Schwarz methods for unstructured spectral elements
Journal of Computational Physics
A finite element multigrid preconditioner for Chebyshev—collocation
Proceedings of the fourth international conference on Spectral and high order methods (ICOSAHOM 1998)
Fast parallel direct solvers for Coarse Grid problems
Journal of Parallel and Distributed Computing
Multigrid Solver for the Inner Problem in Domain Decomposition Methods for p-FEM
SIAM Journal on Numerical Analysis
Spectral Element Methods for Transitional Flows in Complex Geometries
Journal of Scientific Computing
A Schwarz Preconditioner for the Cubed-Sphere
SIAM Journal on Scientific Computing
Algebraic multigrid for higher-order finite elements
Journal of Computational Physics
High-order discontinuous Galerkin methods using an hp-multigrid approach
Journal of Computational Physics
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We study the performance of the multigrid method applied to spectral element (SE) discretizations of the Poisson and Helmholtz equations. Smoothers based on finite element (FE) discretizations, overlapping Schwarz methods, and point-Jacobi are considered in conjunction with conjugate gradient and GMRES acceleration techniques. It is found that Schwarz methods based on restrictions of the originating SE matrices converge faster than FE-based methods and that weighting the Schwarz matrices by the inverse of the diagonal counting matrix is essential to effective Schwarz smoothing. Several of the methods considered achieve convergence rates comparable to those attained by classic multigrid on regular grids.