A fourth-order-accurate Fourier method for the Helmholtz equation in three dimensions
ACM Transactions on Mathematical Software (TOMS)
Tensor product generalized ADI methods for separable elliptic problems
SIAM Journal on Numerical Analysis
A domain-decomposed fast poisson solver on a rectangle
SIAM Journal on Scientific and Statistical Computing - Papers from the Second Conference on Parallel Processing for Scientific Computin
Convergence of O(h4) cubic spline collocation methods for elliptic partial differential equations
SIAM Journal on Numerical Analysis
Algorithm 685: a program for solving separable elliptic equations
ACM Transactions on Mathematical Software (TOMS)
Algebraic spectral multigrid methods
ICOSAHOM '89 Proceedings of the conference on Spectral and high order methods for partial differential equations
Basis-Spline collocation method for the lattice solution of boundary value problems
Journal of Computational Physics
A stabilized treatment of the biharmonic operator with spectral methods
SIAM Journal on Scientific and Statistical Computing
Parallel collocation methods for boundary value problems
Parallel collocation methods for boundary value problems
Computational frameworks for the fast Fourier transform
Computational frameworks for the fast Fourier transform
Fast direct solvers for piecewise Hermite bicubic orthogonal spline collocation equations
SIAM Journal on Numerical Analysis
SIAM Journal on Numerical Analysis
Journal of Computational and Applied Mathematics
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
Algorithm 651: Algorithm HFFT–high-order fast-direct solution of the Helmholtz equation
ACM Transactions on Mathematical Software (TOMS)
On Robbins boundary conditions, elliptic equations, and FTT methods
Journal of Computational Physics
Basis spline collocation method for solving the Schro¨dinger equation in axillary symmetric systems
Journal of Computational Physics
SIAM Journal on Scientific Computing
A new efficient spectral Galerkin method for singular perturbation problems
Journal of Scientific Computing
A Class of Sparse Spectral Operators for Inversion of Powers of the Laplacian in N Dimensions
Journal of Scientific Computing
Parallel implementation of fast elliptic solver
Parallel Computing
Efficient Spectral-Galerkin Methods III: Polar and Cylindrical Geometries
SIAM Journal on Scientific Computing
Spline Collocation Differentiation Matrices
SIAM Journal on Numerical Analysis
An efficient spectral-projection method for the Navier-Stokes equations in cylindrical geometries
Journal of Computational Physics
Efficient iterative solution of the three-dimensional Helmholtz equation
Journal of Computational Physics
SIAM Journal on Scientific Computing
A mixed spectral/wavelet method for the solution of the Stokes problem
Journal of Computational Physics
Galerkin spectral method for the vorticity and stream function equations
Journal of Computational Physics
A Nonstandard Cyclic Reduction Method, Its Variants and Stability
SIAM Journal on Matrix Analysis and Applications
A Fast Direct Solution of Poisson's Equation Using Fourier Analysis
Journal of the ACM (JACM)
A Domain Decomposition Method for the Helmholtz Equation in a Multilayer Domain
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
A Fast Domain Decomposition High Order Poisson Solver
Journal of Scientific Computing
Tensorial basis spline collocation method for Poisson's equation
Journal of Computational Physics
A direct spectral collocation Poisson solver in polar and cylindrical coordinates
Journal of Computational Physics
Galerkin-Legendre spectral method for the 3D Helmholtz equation
Journal of Computational Physics
Orthogonal spline collocation methods for partial differential equations
Journal of Computational and Applied Mathematics - Special issue on numerical analysis 2000 Vol. VII: partial differential equations
An efficient spectral-projection method for the Navier--Stokes equations in cylindrical geometries
Journal of Computational Physics
SIAM Journal on Scientific Computing
A Legendre Spectral Galerkin Method for the Biharmonic Dirichlet Problem
SIAM Journal on Scientific Computing
Discrete-Time Orthogonal Spline Collocation Methods for Vibration Problems
SIAM Journal on Numerical Analysis
Stable and Efficient Spectral Methods in Unbounded Domains Using Laguerre Functions
SIAM Journal on Numerical Analysis
A Kronecker product variant of the FACR method for solving the generalized Poisson equation
Journal of Computational and Applied Mathematics - Special issue: Proceedings of the 9th International Congress on computational and applied mathematics
Accurate ω-ψ spectral solution of the singular driven cavity problem
Journal of Computational Physics
A simple compact fourth-order Poisson solver on polar geometry
Journal of Computational Physics
Separation-of-variables as a preconditioner for an iterative Helmholtz solver
Applied Numerical Mathematics
Matrix Decomposition Algorithms for Modified Spline Collocation for Helmholtz Problems
SIAM Journal on Scientific Computing
Essential imposition of Neumann condition in Galerkin-Legendre elliptic solvers
Journal of Computational Physics
A Nonoverlapping Domain Decomposition Method for Orthogonal Spline Collocation Problems
SIAM Journal on Numerical Analysis
Journal of Computational Physics
Journal of Computational Physics
A Fast Algorithm for the Electromagnetic Scattering from a Large Cavity
SIAM Journal on Scientific Computing
Optimal Superconvergent One Step Nodal Cubic Spline Collocation Methods
SIAM Journal on Scientific Computing
SIAM Journal on Scientific Computing
A Matrix Decomposition MFS Algorithm for Problems in Hollow Axisymmetric Domains
Journal of Scientific Computing
A formally fourth-order accurate compact scheme for 3D Poisson equation in cylindrical coordinates
Journal of Computational and Applied Mathematics
Journal of Computational and Applied Mathematics
A Nonoverlapping Domain Decomposition Method for Legendre Spectral Collocation Problems
Journal of Scientific Computing
An efficient direct parallel spectral-element solver for separable elliptic problems
Journal of Computational Physics
Efficient Chebyshev---Petrov---Galerkin Method for Solving Second-Order Equations
Journal of Scientific Computing
Modified Nodal Cubic Spline Collocation For Poisson's Equation
SIAM Journal on Numerical Analysis
Journal of Computational Physics
A domain decomposition solver for acoustic scattering by elastic objects in layered media
Journal of Computational Physics
A Legendre-Galerkin Spectral Method for Optimal Control Problems Governed by Elliptic Equations
SIAM Journal on Numerical Analysis
Efficient spectral-Galerkin methods for polar and cylindrical geometries
Applied Numerical Mathematics
A Fast Direct Solver for the Biharmonic Problem in a Rectangular Grid
SIAM Journal on Scientific Computing
Efficient multi-dimensional solution of PDEs using Chebyshev spectral methods
Journal of Computational Physics
Fast Tensor-Product Solvers: Partially Deformed Three-dimensional Domains
Journal of Scientific Computing
A Laguerre-Legendre Spectral Method for the Stokes Problem in a Semi-Infinite Channel
SIAM Journal on Numerical Analysis
Journal of Scientific Computing
Efficient MFS Algorithms for Inhomogeneous Polyharmonic Problems
Journal of Scientific Computing
Compact optimal quadratic spline collocation methods for the Helmholtz equation
Journal of Computational Physics
A Fourth Order Hermitian Box-Scheme with Fast Solver for the Poisson Problem in a Square
Journal of Scientific Computing
A Jacobi-Jacobi dual-Petrov-Galerkin method for third- and fifth-order differential equations
Mathematical and Computer Modelling: An International Journal
Computers & Mathematics with Applications
A Fast Preconditioned Iterative Algorithm for the Electromagnetic Scattering from a Large Cavity
Journal of Scientific Computing
Fast poisson solvers for graphics processing units
PARA'12 Proceedings of the 11th international conference on Applied Parallel and Scientific Computing
Efficient MFS Algorithms for Problems in Thermoelasticity
Journal of Scientific Computing
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We provide an overview of matrix decomposition algorithms (MDAs) for the solution of systems of linear equations arising when various discretization techniques are applied in the numerical solution of certain separable elliptic boundary value problems in the unit square. An MDA is a direct method which reduces the algebraic problem to one of solving a set of independent one-dimensional problems which are generally banded, block tridiagonal, or almost block diagonal. Often, fast Fourier transforms (FFTs) can be employed in an MDA with a resulting computational cost of O(N 2 logN) on an N驴脳驴N uniform partition of the unit square. To formulate MDAs, we require knowledge of the eigenvalues and eigenvectors of matrices arising in corresponding two---point boundary value problems in one space dimension. In many important cases, these eigensystems are known explicitly, while in others, they must be computed. The first MDAs were formulated almost fifty years ago, for finite difference methods. Herein, we discuss more recent developments in the formulation and application of MDAs in spline collocation, finite element Galerkin and spectral methods, and the method of fundamental solutions. For ease of exposition, we focus primarily on the Dirichlet problem for Poisson's equation in the unit square, sketch extensions to other boundary conditions and to more involved elliptic problems, including the biharmonic Dirichlet problem, and report extensions to three dimensional problems in a cube. MDAs have also been used extensively as preconditioners in iterative methods for solving linear systems arising from discretizations of non-separable boundary value problems.