Results 1  10
of
39
The Adaptive Multilevel Finite Element Solution of the PoissonBoltzmann Equation on Massively Parallel Computers
 J. COMPUT. CHEM
, 2000
"... Using new methods for the parallel solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element soluti ..."
Abstract

Cited by 43 (14 self)
 Add to MetaCart
Using new methods for the parallel solution of elliptic partial differential equations, the teraflops computing power of massively parallel computers can be leveraged to perform electrostatic calculations on large biological systems. This paper describes the adaptive multilevel finite element solution of the PoissonBoltzmann equation for a microtubule on the NPACI IBM Blue Horizon supercomputer. The microtubule system is 40 nm in length and 24 nm in diameter, consists of roughly 600,000 atoms, and has a net charge of1800 e. PoissonBoltzmann calculations are performed for several processor configurations and the algorithm shows excellent parallel scaling.
Adaptive numerical treatment of elliptic systems on manifolds
 Advances in Computational Mathematics, 15(1):139
, 2001
"... ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element ..."
Abstract

Cited by 40 (24 self)
 Add to MetaCart
ABSTRACT. Adaptive multilevel finite element methods are developed and analyzed for certain elliptic systems arising in geometric analysis and general relativity. This class of nonlinear elliptic systems of tensor equations on manifolds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are considered in some detail. Two a posteriori error indicators are derived, based on local residuals and on global linearized adjoint or dual problems. The design of Manifold Code (MC) is then discussed; MC is an adaptive multilevel finite element software package for 2 and 3manifolds developed over several years at Caltech and UC San Diego. It employs a posteriori error estimation, adaptive simplex subdivision, unstructured algebraic multilevel methods, global inexact Newton methods, and numerical continuation methods for the numerical solution of nonlinear covariant elliptic systems on 2 and 3manifolds. Some of the more interesting features of MC are described in detail, including some new ideas for topology and geometry representation in simplex meshes, and an unusual partition of unitybased method for exploiting parallel computers. A short example is then given which involves the Hamiltonian and momentum constraints in the Einstein equations, a representative nonlinear 4component covariant elliptic system on a Riemannian 3manifold which arises in general relativity. A number of operator properties and solvability results recently established are first summarized, making possible two quasioptimal a priori error estimates for Galerkin approximations which are then derived. These two results complete the theoretical framework for effective use of adaptive multilevel finite element methods. A sample calculation using the MC software is then presented.
The Auxiliary Space Method And Optimal Multigrid Preconditioning Techniques For Unstructured Grids
 Computing
, 1996
"... . An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxi ..."
Abstract

Cited by 31 (2 self)
 Add to MetaCart
. An abstract framework of auxiliary space method is proposed and, as an application, an optimal multigrid technique is developed for general unstructured grids. The auxiliary space method is a (nonnested) two level preconditioning technique based on a simple relaxation scheme (smoother) and an auxiliary space (that may be roughly understood as a nonnested coarser space). An optimal multigrid preconditioner is then obtained for a discretized partial differential operator defined on an unstructured grid by using an auxiliary space defined on a more structured grid in which a further nested multigrid method can be naturally applied. This new technique make it possible to apply multigrid methods to general unstructured grids without too much more programming effort than traditional solution methods. Some simple examples are also given to illustrate the abstract theory and for instance the Morley finite element space is used as an auxiliary space to construct a preconditioner for Argyris ...
Domain decomposition for multiscale PDEs
 Numer. Math
"... We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises of ..."
Abstract

Cited by 30 (14 self)
 Add to MetaCart
We consider additive Schwarz domain decomposition preconditioners for piecewise linear finite element approximations of elliptic PDEs with highly variable coefficients. In contrast to standard analyses, we do not assume that the coefficients can be resolved by a coarse mesh. This situation arises often in practice, for example in the computation of flows in heterogeneous porous media, in both the deterministic and (MonteCarlo simulated) stochastic cases. We consider preconditioners which combine local solves on general overlapping subdomains together with a global solve on a general coarse space of functions on a coarse grid. We perform a new analysis of the preconditioned matrix, which shows rather explicitly how its condition number depends on the variable coefficient in the PDE as well as on the coarse mesh and overlap parameters. The classical estimates for this preconditioner with linear coarsening guarantee good conditioning only when the coefficient varies mildly inside the coarse grid elements. By contrast, our new results show that, with a good choice of subdomains and coarse space basis functions, the preconditioner can still be robust even for large coefficient variation inside domains,
Additive Schwarz Domain Decomposition Methods For Elliptic Problems On Unstructured Meshes
 Numerical Algorithms
, 1994
"... . We give several additive Schwarz domain decomposition methods for solving finite element problems which arise from the discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Our theory requires no assumption (for the main results) on the substructures whic ..."
Abstract

Cited by 27 (13 self)
 Add to MetaCart
. We give several additive Schwarz domain decomposition methods for solving finite element problems which arise from the discretizations of elliptic problems on general unstructured meshes in two and three dimensions. Our theory requires no assumption (for the main results) on the substructures which constitute the whole domain, so each substructure can be of arbitrary shape and of different size. The global coarse mesh is allowed to be nonnested to the fine grid on which the discrete problem is to be solved and both the coarse meshes and the fine meshes need not be quasiuniform. In this general setting, our algorithms have the same optimal convergence rate of the usual domain decomposition methods on structured meshes. The condition numbers of the preconditoned systems depend only on the (possibly small) overlap of the substructures and the size of the coarse grid, but is independent of the sizes of the subdomains. Key Words. Unstructured meshes, nonnested coarse meshes, additive ...
THE USE OF POINTWISE INTERPOLATION IN DOMAIN DECOMPOSITION METHODS WITH NONNESTED MESHES
, 1995
"... In this paper, we develop a new technique, and a corresponding theory, for Schwarz type overlapping domain decomposition methods for solving large sparse linear systems which arise from nite element discretization of elliptic partial di erential equations. The theory provides an optimal convergence ..."
Abstract

Cited by 23 (3 self)
 Add to MetaCart
In this paper, we develop a new technique, and a corresponding theory, for Schwarz type overlapping domain decomposition methods for solving large sparse linear systems which arise from nite element discretization of elliptic partial di erential equations. The theory provides an optimal convergence of an additive Schwarz algorithm that is constructed with a nonnested coarse space, and a not necessarily shape regular subdomain partitioning. The theory is also applicable to the graph partitioning algorithms recently developed, [5, 15], for problems de ned on unstructured meshes.
Parallel Computation of Flow in Heterogeneous Media Modelled by Mixed Finite Elements
"... In this paper we describe a fast parallel method for highly illconditioned saddlepoint systems arising from mixed finite element simulations of stochastic partial differential equations (PDEs) modelling flow in heterogeneous media. Each realisation of these stochastic PDEs requires the solution of ..."
Abstract

Cited by 17 (14 self)
 Add to MetaCart
In this paper we describe a fast parallel method for highly illconditioned saddlepoint systems arising from mixed finite element simulations of stochastic partial differential equations (PDEs) modelling flow in heterogeneous media. Each realisation of these stochastic PDEs requires the solution of the linear firstorder velocitypressure system comprising Darcy's law coupled with an incompressibility constraint. The chief difficulty is that the permeability may be highly variable, especially when the statistical model has a large variance and a small correlation length. For reasonable accuracy, the discretisation has to be extremely fine. We solve these problems by first reducing the saddlepoint formulation to a symmetric positive definite (SPD) problem using a suitable basis for the space of divergencefree velocities. The reduced problem is solved using parallel conjugate gradients preconditioned with an algebraically determined additive Schwarz domain decomposition preconditioner. The resu...
A Domain Decomposition Preconditioner for a Parallel Finite Element Solver on Distributed Unstructured Grids
 Parallel Computing
, 1995
"... We consider a number of practical issues associated with the parallel distributed memory solution of elliptic partial differential equations using unstructured meshes in two dimensions. The first part of the paper describes a parallel mesh generation algorithm which is designed both for efficiency a ..."
Abstract

Cited by 15 (11 self)
 Add to MetaCart
We consider a number of practical issues associated with the parallel distributed memory solution of elliptic partial differential equations using unstructured meshes in two dimensions. The first part of the paper describes a parallel mesh generation algorithm which is designed both for efficiency and to produce a wellpartitioned, distributed mesh, suitable for the efficient parallel solution of an elliptic p.d.e. The second part of the paper concentrates on parallel domain decomposition preconditioning for the linear algebra problems which arise when solving such a p.d.e. on the unstructured meshes that we generate. It is demonstrated that by allowing the mesh generator and the p.d.e. solver to share a certain coarse grid structure we are able to obtain efficient parallel solutions to a number of large problems. Although the work is presented here in a finite element context, the issues of mesh generation and domain decomposition are not of course strictly dependent upon this particu...
Numerical Identifications of Parameters in Parabolic Systems
 Inverse Problems
, 1998
"... . In this paper, we investigate the numerical identifications of physical parameters in parabolic initialboundary value problems. The identifying problem is first formulated as a constrained minimization one using the output least squares approach with the H 1 regularization or BV regularizatio ..."
Abstract

Cited by 14 (9 self)
 Add to MetaCart
. In this paper, we investigate the numerical identifications of physical parameters in parabolic initialboundary value problems. The identifying problem is first formulated as a constrained minimization one using the output least squares approach with the H 1 regularization or BV regularization. Then a simple finite element method is used to approximate the constrained minimization problem and the convergence of the approximation is shown for both regularizations. The discrete constrained problem can be reduced to a sequence of unconstrained minimization problems. Numerical experiments are presented to show the efficiency of the proposed method, even for identifying highly discontinuous and oscillating parameters. 1. Introduction In this paper, we consider a finite element approach, combined with the output least squares method, for identifying the parameter q(x) in the following parabolic problem #u #t  # (q(x)#u) = f (x, t ) in # (0, T ) (1.1) with the initial condition ...
Boundary Treatments For Multilevel Methods On Unstructured Meshes
, 1996
"... . In applying multilevel iterative methods on unstructured meshes, the grid hierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid. In this case, the standard coarsetofine grid transfer operators with linear interpolants are not accurate enoug ..."
Abstract

Cited by 14 (8 self)
 Add to MetaCart
. In applying multilevel iterative methods on unstructured meshes, the grid hierarchy can allow general coarse grids whose boundaries may be nonmatching to the boundary of the fine grid. In this case, the standard coarsetofine grid transfer operators with linear interpolants are not accurate enough at Neumann boundaries so special care is needed to correctly handle di#erent types of boundary conditions. We propose two e#ective ways to adapt the standard coarsetofine interpolations to correctly implement boundary conditions for twodimensional polygonal domains, and we provide some numerical examples of multilevel Schwarz methods (and multigrid methods) which show that these methods are as e#cient as in the structured case. In addition, we prove that the proposed interpolants possess the local optimal L 2 approximation and H 1 stability, which are essential in the convergence analysis of the multilevel Schwarz methods. Using these results, we give a condition number bound for ...