On the eigenvalue distribution of a class of preconditioning methods
Numerische Mathematik
High performance preconditioning
SIAM Journal on Scientific and Statistical Computing
Finite-element preconditioning for pseudospectral solutions of elliptic problems
SIAM Journal on Scientific and Statistical Computing
SIAM Journal on Numerical Analysis
Preconditioning Legendre spectral collocation approximations to elliptic problems
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Quasi-Optimal Schwarz Methods for the Conforming Spectral Element Discretization
SIAM Journal on Numerical Analysis
A combined unifrontal/multifrontal method for unsymmetric sparse matrices
ACM Transactions on Mathematical Software (TOMS)
The Multifrontal Solution of Indefinite Sparse Symmetric Linear
ACM Transactions on Mathematical Software (TOMS)
Multigrid Solver for the Inner Problem in Domain Decomposition Methods for p-FEM
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
On condition numbers in hp-FEM with Gauss-Lobatto-based shape functions
Journal of Computational and Applied Mathematics
MA57---a code for the solution of sparse symmetric definite and indefinite systems
ACM Transactions on Mathematical Software (TOMS)
Multiresolution weighted norm equivalences and applications
Numerische Mathematik
Strategies for Scaling and Pivoting for Sparse Symmetric Indefinite Problems
SIAM Journal on Matrix Analysis and Applications
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Several old and new finite-element preconditioners for nodal-based spectral discretizations of $-\Delta u=f$ in the domain $\Omega=(-1,1)^d$ ($d=2$ or 3), with Dirichlet or Neumann boundary conditions, are considered and compared in terms of both condition number and computational efficiency. The computational domain covers the case of classical single-domain spectral approximations (see [C. Canuto et al., Spectral Methods. Fundamentals in Single Domains, Springer, Heidelberg, 2006]), as well as that of more general spectral-element methods in which the preconditioners are expressed in terms of local (upon every element) algebraic solvers. The primal spectral approximation is based on the Galerkin approach with numerical integration (G-NI) at the Legendre-Gauss-Lobatto (LGL) nodes in the domain. The preconditioning matrices rely on either $\mathbb{P}_1$, $\mathbb{Q}_1$, or $\mathbb{Q}_{1,NI}$ (i.e., with numerical integration) finite elements on meshes whose vertices coincide with the LGL nodes used for the spectral approximation. The analysis highlights certain preconditioners, which yield the solution at an overall cost proportional to $N^{d+1}$, where $N$ denotes the polynomial degree in each direction.