Results 1 
6 of
6
SOME FAST ALGORITHMS FOR HIERARCHICALLY SEMISEPARABLE MATRICES
"... Abstract. In this paper we generalize the hierarchically semiseparable (HSS) representations and propose some fast algorithms for HSS matrices. We provide a new linear complexity ULV T factorization algorithm for symmetric positive definite HSS matrices with small offdiagonal ranks. The correspondi ..."
Abstract

Cited by 48 (4 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we generalize the hierarchically semiseparable (HSS) representations and propose some fast algorithms for HSS matrices. We provide a new linear complexity ULV T factorization algorithm for symmetric positive definite HSS matrices with small offdiagonal ranks. The corresponding factors can be used to solve compact HSS systems also in linear complexity. Numerical examples demonstrate the efficiency of the solver. We also present fast algorithms including new HSS structure generation, HSS form Cholesky factorization, and model compression. These algorithms are useful for problems where offdiagonal blocks have small numerical ranks. Key words. HSS matrix, fast algorithms, generalized HSS Cholesky factorization AMS subject classifications. 65F05 1. Introduction. In
MULTILEVEL PRECONDITIONERS FOR DISCONTINUOUS GALERKIN APPROXIMATIONS OF ELLIPTIC PROBLEMS WITH JUMP COEFFICIENTS
, 2010
"... In this article we develop and analyze twolevel and multilevel methods for the family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with rough coefficients (exhibiting large jumps across interfaces in the domain). These methods are based on ..."
Abstract

Cited by 12 (5 self)
 Add to MetaCart
(Show Context)
In this article we develop and analyze twolevel and multilevel methods for the family of Interior Penalty (IP) discontinuous Galerkin (DG) discretizations of second order elliptic problems with rough coefficients (exhibiting large jumps across interfaces in the domain). These methods are based on a decomposition of the DG finite element space that inherently hinges on the diffusion coefficient of the problem. Our analysis of the proposed preconditioners is presented for both symmetric and nonsymmetric IP schemes, and we establish both robustness with respect to the jump in the coefficient and nearoptimality with respect to the mesh size. Following the analysis, we present a sequence of detailed numerical results which verify the
Adaptive Finite Element Methods with Inexact Solvers for the Nonlinear PoissonBoltzmann Equation
, 1107
"... In this article we study adaptive finite element methods (AFEM) with inexact solvers for a class of semilinear elliptic interface problems. We are particularly interested in nonlinear problems with discontinuous diffusion coefficients, such as the nonlinear PoissonBoltzmann equation and its regular ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
(Show Context)
In this article we study adaptive finite element methods (AFEM) with inexact solvers for a class of semilinear elliptic interface problems. We are particularly interested in nonlinear problems with discontinuous diffusion coefficients, such as the nonlinear PoissonBoltzmann equation and its regularizations. The algorithm we study consists
OPTIMAL MULTILEVEL METHODS FOR GRADED BISECTION GRIDS
"... We design and analyze optimal additive and multiplicative multilevel methods for solving H 1 problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs to be pe ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
We design and analyze optimal additive and multiplicative multilevel methods for solving H 1 problems on graded grids obtained by bisection. We deal with economical local smoothers: after a global smoothing in the finest mesh, local smoothing for each added node during the refinement needs to be performed only for three vertices the new vertex and its two parent vertices. We show that our methods lead to optimal complexity for any dimensions and polynomial degree. The theory hinges on a new decomposition of bisection grids in any dimension, which is of independent interest and yields a corresponding decomposition of spaces. We use the latter to bridge the gap between graded and quasiuniform grids, for which the multilevel theory is wellestablished.
MULTILEVEL PRECONDITIONERS FOR REACTIONDIFFUSION PROBLEMS WITH DISCONTINUOUS COEFFICIENTS
"... ABSTRACT. In this paper, we extend some of the multilevel convergence results obtained by Xu and Zhu in [Xu and Zhu, M3AS 2008], to the case of second order linear reactiondiffusion equations. Specifically, we consider the multilevel preconditioners for solving the linear systems arising from the ..."
Abstract
 Add to MetaCart
(Show Context)
ABSTRACT. In this paper, we extend some of the multilevel convergence results obtained by Xu and Zhu in [Xu and Zhu, M3AS 2008], to the case of second order linear reactiondiffusion equations. Specifically, we consider the multilevel preconditioners for solving the linear systems arising from the linear finite element approximation of the problem, where both diffusion and reaction coefficients are piecewiseconstant functions. We discuss in detail the influence of both the discontinuous reaction and diffusion coefficients to the performance of the classical BPX and multigrid Vcycle preconditioner. 1.
Unstructured Geometric Multigrid in Two and Three Dimensions on Complex and Graded Meshes
, 2013
"... The use of multigrid and related preconditioners with the finite element method is often limited by the difficulty of applying the algorithm effectively to a problem, especially when the domain has a complex shape or the mesh has adaptive refinement. We introduce a simplification of a general topolo ..."
Abstract
 Add to MetaCart
(Show Context)
The use of multigrid and related preconditioners with the finite element method is often limited by the difficulty of applying the algorithm effectively to a problem, especially when the domain has a complex shape or the mesh has adaptive refinement. We introduce a simplification of a general topologicallymotivated mesh coarsening algorithm for use in creating hierarchies of meshes for geometric unstructured multigrid methods. The connections between the guarantees of this technique and the quality criteria necessary for multigrid methods for nonquasiuniform problems are noted. The implementation details, in particular those related to coarsening, remeshing, and interpolation, are discussed. Computational tests on pathological test cases from adaptive finite element methods show the performance of the technique. 1