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A review of algebraic multigrid
, 2001
"... Since the early 1990s, there has been a strongly increasing demand for more efficient methods to solve large sparse, unstructured linear systems of equations. For practically relevant problem sizes, classical onelevel methods had already reached their limits and new hierarchical algorithms had to b ..."
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Cited by 347 (11 self)
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to be developed in order to allow an efficient solution of even larger problems. This paper gives a review of the first hierarchical and purely matrixbased approach, algebraic multigrid (AMG). AMG can directly be applied, for instance, to efficiently solve various types of elliptic partial differential equations
The Immersed Interface Method for Elliptic Equations with Discontinuous Coefficients and Singular Sources
 SIAM J. Num. Anal
, 1994
"... Abstract. The authors develop finite difference methods for elliptic equations of the form V. ((x)Vu(x)) + (x)u(x) f(x) in a region in one or two space dimensions. It is assumed that gt is a simple region (e.g., a rectangle) and that a uniform rectangular grid is used. The situation is studied in wh ..."
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Cited by 273 (31 self)
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itself across F is allowed. It is shown that it is possible to modify the standard centered difference approximation to maintain second order accuracy on the uniform grid even when F is not aligned with the grid. This approach is also compared with a discrete delta function approach to handling singular
Elliptic and parabolic secondorder PDEs with growing coefficients
"... Abstract. We consider a secondorder parabolic equation in R d+1 with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even in this ca ..."
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Cited by 18 (4 self)
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Abstract. We consider a secondorder parabolic equation in R d+1 with possibly unbounded lower order coefficients. All coefficients are assumed to be only measurable in the time variable and locally Hölder continuous in the space variables. We show that global Schauder estimates hold even
ImplicitExplicit Methods For TimeDependent PDEs
 SIAM J. NUMER. ANAL
, 1997
"... Implicitexplicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusionconvection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used for the convection ..."
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Cited by 178 (6 self)
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Implicitexplicit (IMEX) schemes have been widely used, especially in conjunction with spectral methods, for the time integration of spatially discretized PDEs of diffusionconvection type. Typically, an implicit scheme is used for the diffusion term and an explicit scheme is used
Numerical Linear Algebra with Applications, Vol. 4(5), 369–391 (1997) A Nearly Optimal Preconditioning based on Recursive Red–black Orderings
"... Considering matrices obtained by the application of a fivepoint stencil on a 2D rectangular grid, we analyse a preconditioning method introduced by Axelsson and Eijkhout, and by Brand and Heinemann. In this method, one performs a (modified) incomplete factorization with respect to a socalled ‘repe ..."
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called ‘repeated ’ or ‘recursive ’ red– black ordering of the unknowns while fillin is accepted provided that the red unknowns in a same level remain uncoupled. Considering discrete second order elliptic PDEs with isotropic coefficients, we show that the condition number is bounded by �(n 1 2 log2 (√5−1) ) where
Quadratic Spline Collocation Methods for Systems of Elliptic PDEs
, 2001
"... We consider Quadratic Spline Collocation (QSC) methods for solving systems of two coupled linear secondorder PDEs in two dimensions. The system of PDEs is treated as one entity; no decoupling is applied. Optimal order approximation to the solution is obtained, in the sense that the convergence orde ..."
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Cited by 4 (2 self)
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We consider Quadratic Spline Collocation (QSC) methods for solving systems of two coupled linear secondorder PDEs in two dimensions. The system of PDEs is treated as one entity; no decoupling is applied. Optimal order approximation to the solution is obtained, in the sense that the convergence
Discretization and Solution of Elliptic PDEsA Digital Signal Processing Approach
"... A digital signal processing (DSP) approach is used to study numerical methods for discretizing and solving linear elliptic partial differential equations (PDEs). Whereas conventional PDE analysis techniques rely on matrix analysis and on a spacedomain point of view to study the performance of sol ..."
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A digital signal processing (DSP) approach is used to study numerical methods for discretizing and solving linear elliptic partial differential equations (PDEs). Whereas conventional PDE analysis techniques rely on matrix analysis and on a spacedomain point of view to study the performance
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS Numer. Linear Algebra Appl., 5, 441–459 (1998) Optimal Vcycle Algebraic Multilevel Preconditioning
"... We consider algebraic multilevel preconditioning methods based on the recursive use of a 2 × 2 block incomplete factorization procedure in which the Schur complement is approximated by a coarse grid matrix. As is well known, for discrete secondorder elliptic PDEs, optimal convergence properties are ..."
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We consider algebraic multilevel preconditioning methods based on the recursive use of a 2 × 2 block incomplete factorization procedure in which the Schur complement is approximated by a coarse grid matrix. As is well known, for discrete secondorder elliptic PDEs, optimal convergence properties
BoseEinstein Condensation in a Gas of Sodium Atoms
, 1995
"... We have observed BoseEinstein condensation of sodium atoms. The atoms were trapped in a novel trap that employed both magnetic and optical forces. Evaporative cooling increased the phasespace density by 6 orders of magnitude within seven seconds. Condensates contained up to 5 3 10 5 atoms at dens ..."
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Cited by 282 (6 self)
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We have observed BoseEinstein condensation of sodium atoms. The atoms were trapped in a novel trap that employed both magnetic and optical forces. Evaporative cooling increased the phasespace density by 6 orders of magnitude within seven seconds. Condensates contained up to 5 3 10 5 atoms
Application of BiCGSTAB to discretized coupled PDEs
"... Aversion of the BiCGSTAB method is applied for solving the linear systems that typically occur when applying the Newton method to a discretized set of coupled elliptic partial di#erential equations in two dimensions. The Incomplete Line LU decomposition is generalized for the case of coupled equ ..."
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set of coupled second order elliptic partial di#erential equations #PDEs# in two dimensions....
Results 11  20
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