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37
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 ..."
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Cited by 89 (17 self)
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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 ..."
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Cited by 57 (26 self)
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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.
A New Paradigm for Parallel Adaptive Meshing Algorithms
 SIAM J. Sci. Comput
, 2003
"... We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to ru ..."
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Cited by 46 (9 self)
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We present a new approach to the use of parallel computers with adaptive finite element methods. This approach addresses the load balancing problem in a new way, requiring far less communication than current approaches. It also allows existing sequential adaptive PDE codes such as PLTMG and MC to run in a parallel environment without a large investment in recoding. In this new approach, the load balancing problem is reduced to the numerical solution of a small elliptic problem on a single processor, using a sequential adaptive solver, without requiring any modifications to the sequential solver. The small elliptic problem is used to produce a posteriori error estimates to predict future element densities in the mesh, which are then used in a weighted recursive spectral bisection of the initial mesh. The bulk of the calculation then takes place independently on each processor, with no communication, using possibly the same sequential adaptive solver. Each processor adapts its region of the mesh independently, and a nearly loadbalanced mesh distribution is usually obtained as a result of the initial weighted spectral bisection. Only the initial fanout of the mesh decomposition to the processors requires communication. Two additional steps requiring boundary exchange communication may be employed after the individual processors reach an adapted solution, namely, the construction of a global conforming mesh from the independent subproblems, followed by a final smoothing phase using the subdomain solutions as an initial guess. We present a series of convincing numerical experiments that illustrate the e#ectiveness of this approach. The justification of the initial refinement prediction step, as well as the justification of skipping the two communicationintensive steps, ...
A twogrid discretization scheme for eigenvalue problems
 MATH. COMP
, 1999
"... A twogrid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid, a ..."
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Cited by 23 (6 self)
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A twogrid discretization scheme is proposed for solving eigenvalue problems, including both partial differential equations and integral equations. With this new scheme, the solution of an eigenvalue problem on a fine grid is reduced to the solution of an eigenvalue problem on a much coarser grid, and the solution of a linear algebraic system on the fine grid and the resulting solution still maintains an asymptotically optimal accuracy.
Generalized Green's Functions and the Effective Domain of Influence
 SIAM J. Sci. Comput
, 2002
"... One wellknown approach to a posteriori analysis of finite element solutions of elliptic problems estimates the error in a quantity of interest in terms of residuals and a generalized Green's function. The generalized Green's function solves the adjoint problem with data related to a quant ..."
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Cited by 18 (7 self)
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One wellknown approach to a posteriori analysis of finite element solutions of elliptic problems estimates the error in a quantity of interest in terms of residuals and a generalized Green's function. The generalized Green's function solves the adjoint problem with data related to a quantity of interest and measures the e#ects of stability, including any decay of influence characteristic of elliptic problems. We show that consideration of the generalized Green's function can be used to improve the e#ciency of the solution process when the goal is to compute multiple quantities of interest and/or to compute quantities of interest that involve globallysupported information such as average values and norms. In the latter case, we introduce a solution decomposition in which we solve a set of problems involving localized information, and then recover the desired information by combining the local solutions. By treating each computation of a quantity of interest independently, the maximum number of elements required to achieve the desired accuracy can be decreased significantly.
Applications of Domain Decomposition and Partition of Unity Methods in Physics and Geometry
, 2003
"... In this article we consider a class of adaptive multilevel domain decompositionlike algorithms, built from a combination of adaptive multilevel finite element, domain decomposition, and partition of unity methods. These algorithms have several interesting features such as very low communication req ..."
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Cited by 10 (4 self)
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In this article we consider a class of adaptive multilevel domain decompositionlike algorithms, built from a combination of adaptive multilevel finite element, domain decomposition, and partition of unity methods. These algorithms have several interesting features such as very low communication requirements, and they inherit a simple and elegant approximation theory framework from partition of unity methods. They are also very easy to use with highly complex sequential adaptive finite element packages, requiring little or no modification of the underlying sequential finite element software. The parallel algorithm can be implemented as a simple loop which starts o# a sequential local adaptive solve on a collection of processors simultaneously
A Short Course on Duality, Adjoint Operators, Green’s Functions, and A Posteriori Error Analysis
, 2004
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Finite element approximations for Schrödinger equations with applications to electronic structure computations
 J. Comput. Math
"... Dedicated to the 70th birthday of Professor Junzhi Cui In this paper, both the standard finite element discretization and a twoscale finite element discretization for Schrödinger equations are studied. The numerical analysis is based on the regularity that is also obtained in this paper for the Sc ..."
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Cited by 6 (5 self)
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Dedicated to the 70th birthday of Professor Junzhi Cui In this paper, both the standard finite element discretization and a twoscale finite element discretization for Schrödinger equations are studied. The numerical analysis is based on the regularity that is also obtained in this paper for the Schrödinger equations. Very satisfying applications to electronic structure computations are provided, too.
FINITE ELEMENT ERROR ESTIMATES FOR CRITICAL GROWTH SEMILINEAR PROBLEMS WITHOUT ANGLE CONDITIONS
, 1108
"... ABSTRACT. In this article we consider a priori error and pointwise estimates for finite element approximations of solutions to semilinear elliptic boundary value problems in d � 2 space dimensions, with nonlinearities satisfying critical growth conditions. It is wellunderstood how mesh geometry imp ..."
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Cited by 5 (5 self)
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ABSTRACT. In this article we consider a priori error and pointwise estimates for finite element approximations of solutions to semilinear elliptic boundary value problems in d � 2 space dimensions, with nonlinearities satisfying critical growth conditions. It is wellunderstood how mesh geometry impacts finite element interpolant quality, and leads to the reasonable notion of shape regular simplex meshes. It is also wellknown how to perform both mesh generation and simplex subdivision, in arbitrary space dimension, so as to guarantee the entire hierarchy of nested simplex meshes produced through subdivision continue to satisfy shape regularity. However, much more restrictive angle conditions are needed for basic a priori quasioptimal error estimates, as well as for a priori pointwise estimates. These angle conditions, which are particularly difficult to satisfy in three dimensions in any type of unstructured or adaptive setting, are needed to gain pointwise control of the nonlinearity through discrete maximum principles. This represents a major gap in finite element approximation theory for nonlinear problems on unstructured meshes, and in particular for adaptive methods. In this article, we close this gap in the case of semilinear problems with critical or subcritical nonlinear growth, by
Local and Parallel Finite Element Algorithms for the Stokes Problem ∗
, 2006
"... Based on twogrid discretizations, some new local and parallel finite element algorithms for the Stokes problem are proposed and analyzed in this paper. These algorithms are motivated by the observation that for a solution to the Stokes problem, low frequency components can be approximated well by a ..."
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Cited by 5 (0 self)
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Based on twogrid discretizations, some new local and parallel finite element algorithms for the Stokes problem are proposed and analyzed in this paper. These algorithms are motivated by the observation that for a solution to the Stokes problem, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on a fine grid by some local and parallel procedure. One technical tool for the analysis is some local a priori estimates that are also obtained in this paper for the finite element solutions on general shaperegular grids.