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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 88 (16 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.
Hierarchical Bases and the Finite Element Method
, 1997
"... CONTENTS 1 Introduction 1 2 Preliminaries 3 3 Fundamental TwoLevel Estimates 7 4 A Posteriori Error Estimates 16 5 TwoLevel Iterative Methods 23 6 Multilevel Cauchy Inequalities 30 7 Multilevel Iterative Methods 34 References 41 1. Introduction In this work we present a brief introduction to hie ..."
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Cited by 79 (4 self)
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CONTENTS 1 Introduction 1 2 Preliminaries 3 3 Fundamental TwoLevel Estimates 7 4 A Posteriori Error Estimates 16 5 TwoLevel Iterative Methods 23 6 Multilevel Cauchy Inequalities 30 7 Multilevel Iterative Methods 34 References 41 1. Introduction In this work we present a brief introduction to hierarchical bases, and the important part they play in contemporary finite element calculations. In particular, we examine their role in a posteriori error estimation, and in the Department of Mathematics, University of California at San Diego, La Jolla, CA 92093. The work of this author was supported by the Office of Naval Research under contract N0001489J1440. 2 Randolph E. Bank formulation of iterative methods for solving the large sparse sets of linear equations arising from the finite element discretization. Our goal is that the development should be largely selfcontained, but at the same time accessible and interest
GoalOriented Error Estimation and Adaptivity for the Finite Element Method
 COMPUT. MATH. APPL
, 1999
"... In this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These socalled quantities of interest are characterized by linear functionals ..."
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Cited by 75 (8 self)
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In this paper, we study a new approach in a posteriori error estimation, in which the numerical error of finite element approximations is estimated in terms of quantities of interest rather than the classical energy norm. These socalled quantities of interest are characterized by linear functionals on the space of functions to where the solution belongs. We present here the theory with respect to a class of elliptic boundaryvalue problems, and in particular, show how to obtain accurate estimates as well as upper and lowerbounds on the error. We also study the new concept of goaloriented adaptivity, which embodies mesh adaptation procedures designed to control error in specific quantities. Numerical experiments confirm that such procedures greatly accelerate the attainment of local features of the solution to preset accuracies as compared to traditional adaptive schemes based on energy norm error estimates.
Mesh Smoothing Using A Posteriori Error Estimates
 SIAM JOURNAL ON NUMERICAL ANALYSIS
, 1997
"... We develop a simple mesh smoothing algorithm for adaptively improving finite element triangulations. The algorithm makes use of a posteriori error estimates which are now widely used in finite element calculations. In this paper, we derive the method, present some numerical illustrations, and give a ..."
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Cited by 67 (2 self)
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We develop a simple mesh smoothing algorithm for adaptively improving finite element triangulations. The algorithm makes use of a posteriori error estimates which are now widely used in finite element calculations. In this paper, we derive the method, present some numerical illustrations, and give a brief analysis of the issue of uniqueness.
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 55 (25 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.
Adaptive Isogeometric Analysis by Local hRefinement with TSplines
"... Isogeometric analysis based on NURBS (NonUniform Rational BSplines) as basis functions preserves the exact geometry but suffers from the drawback of a rectangular grid of control points in the parameter space, which renders a purely local refinement impossible. This paper demonstrates how this dif ..."
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Cited by 50 (5 self)
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Isogeometric analysis based on NURBS (NonUniform Rational BSplines) as basis functions preserves the exact geometry but suffers from the drawback of a rectangular grid of control points in the parameter space, which renders a purely local refinement impossible. This paper demonstrates how this difficulty can be overcome by using Tsplines instead. Tsplines allow the introduction of socalled Tjunctions, which are related to hanging nodes in the standard FEM. Obeying a few straightforward rules, rectangular patches in the parameter space of the Tsplines can be subdivided and thus a local refinement becomes feasible while still preserving the exact geometry. Furthermore, it is shown how stateoftheart a posteriori error estimation techniques can be combined with refinement by TSplines. Numerical examples underline the potential of isogeometric analysis with Tsplines and give hints for further developments. Key words: adaptivity, a posteriori error estimation, isogeometric analysis, NURBS, CAD,
A Posteriori Error Estimates for Elliptic Problems in Two and Three Space Dimensions
 SIAM J. NUMER. ANAL
, 1993
"... Let u 2 H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error jj u \Gamma ~ u jj, measuring the (local) quality of ~ u, play a cruci ..."
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Cited by 49 (8 self)
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Let u 2 H be the exact solution of a given selfadjoint elliptic boundary value problem, which is approximated by some ~ u 2 S, S being a suitable finite element space. Efficient and reliable a posteriori estimates of the error jj u \Gamma ~ u jj, measuring the (local) quality of ~ u, play a crucial role in termination criteria and in the adaptive refinement of the underlying mesh. A wellknown class of error estimates can be derived systematically by localizing the discretized defect problem using domain decomposition techniques. In the present paper, we provide a guideline for the theoretical analysis of such error estimates. We further clarify the relation to other concepts. Our analysis leads to new error estimates, which are specially suited to three space dimensions. The theoretical results are illustrated by numerical computations.
Anisotropic Mesh Transformations and Optimal Error Control
 Applied Numer. Math
, 1992
"... Recently, research originating in several different applications has appeared on unstructured triangular meshes in which the vertex distribution is not locally uniform, i.e. anisotropic unstructured meshes. The techniques used have the common features that the distribution of triangle shapes for ..."
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Cited by 48 (3 self)
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Recently, research originating in several different applications has appeared on unstructured triangular meshes in which the vertex distribution is not locally uniform, i.e. anisotropic unstructured meshes. The techniques used have the common features that the distribution of triangle shapes for the mesh is controlled by specifying a symmetric tensor, and that the anisotropic mesh is the transform of an isotropic mesh. We discuss how these mechanisms arise in the theory of optimal error control, using simple model mesh generation problems, and review the related research in applications to computational fluid dynamics, surface triangulation, and semiconductor simulation. 1 Introduction In order to fully exploit the flexibility of triangular meshes in the plane, it would intuitively seem likely that it would be necessary to take advantage of the independence of the two length scales of the triangles, e.g. the length of the longest edge, and the perpendicular distance to the opp...
Local and parallel finite element algorithms based on twogrid discretizations
 Math. Comput
"... Abstract. A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, low frequency components ..."
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Cited by 37 (13 self)
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Abstract. A number of new local and parallel discretization and adaptive finite element algorithms are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic problems, 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. The theoretical tools for analyzing these methods are some local a priori and a posteriori estimates that are also obtained in this paper for finite element solutions on general shaperegular grids. Some numerical experiments are also presented to support the theory. 1.
Hierarchical A Posteriori Error Estimators For Mortar Finite Element Methods With Lagrange Multipliers
 SIAM J. Numer. Anal
"... . Hierarchical a posteriori error estimators are introduced and analyzed for mortar finite element methods. A weak continuity condition at the interfaces is enforced by means of Lagrange multipliers. The two proposed error estimators are based on a defect correction in higher order finite element sp ..."
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Cited by 25 (7 self)
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. Hierarchical a posteriori error estimators are introduced and analyzed for mortar finite element methods. A weak continuity condition at the interfaces is enforced by means of Lagrange multipliers. The two proposed error estimators are based on a defect correction in higher order finite element spaces and an adequate hierarchical twolevel splitting. The first provides upper and lower bounds for the discrete energy norm of the mortar finite element solution whereas the second also estimates the error for the Lagrange multiplier. It is shown that an appropriate measure for the nonconformity of the mortar finite element solution is the weighted L 2 norm of the jumps across the interfaces. Key words. mortar finite elements, Lagrange multiplier, mesh dependent norms, a posteriori error estimation, adaptive grid refinement AMS subject classifications. 65N15, 65N30, 65N50, 65N55 1. Introduction. In this paper, we consider a special nonoverlapping domain decomposition method for the di...