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71
A FeedBack Approach to Error Control in Finite Element Methods: Basic Analysis and Examples
 EastWest J. Numer. Math
, 1996
"... this paper. ..."
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 Review of A Posteriori Error Estimation
 and Adaptive MeshRefinement Techniques, Wiley & Teubner
, 1996
"... linear parabolic equations ..."
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A posteriori error estimate for the mixed finite element method
 Math. Comp
, 1997
"... Abstract. A computable error bound for mixed finite element methods is established in the model case of the Poisson–problem to control the error in the H(div,Ω) ×L 2 (Ω)–norm. The reliable and efficient a posteriori error estimate applies, e.g., to Raviart–Thomas, BrezziDouglasMarini, and BrezziD ..."
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Cited by 47 (8 self)
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Abstract. A computable error bound for mixed finite element methods is established in the model case of the Poisson–problem to control the error in the H(div,Ω) ×L 2 (Ω)–norm. The reliable and efficient a posteriori error estimate applies, e.g., to Raviart–Thomas, BrezziDouglasMarini, and BrezziDouglasFortinMarini elements. 1. Mixed method for the Poisson problem Mixed finite element methods are wellestablished in the numerical treatment of partial differential equations as regards a priori error estimates to guarantee convergence [BF]. In practical applications, a posteriori error control is at least of the same importance to guarantee a reliable approximation. Moreover, a posteriori error estimators indicate adaptive meshrefinement criteria [EEHJ, V1] for an efficient computation. In this paper we establish an efficient and reliable error estimator for the model example in the mixed finite element methods: Given f ∈ L2 (Ω), the Poisson problem consists in finding a function u ∈ H1 0 (Ω) that satisfies (1.1) div(A∇u)+f =0 inΩ. Here, A ∈ L ∞ (Ω; R2×2) is symmetric and uniformly elliptic, Ω is a convex bounded domain in the plane with polygonal boundary Γ. The Lebesgue and Sobolev spaces (Ω) are defined as usual (e.g., as in [H, LM]). We assume below that L2 (Ω) and H1 0 (1.1) is H2 –regular which, according to Ω being convex, means certain regularity on A (A the unit matrix as for the Laplace equation is clearly sufficient). The mixed formulation is given by splitting (1.1) into two equations where u ∈ H1 0 (Ω) and p ∈ L2 (Ω) 2 are unknown and have to satisfy (1.2) div p + f =0 and p=A∇u in Ω. It is wellknown that (1.2) has a solution (p, u) ∈ H(div,Ω)×L 2 (Ω), where, as usual, H(div,Ω): = {q ∈ L 2 (Ω) 2:divq∈L 2 (Ω)} is endowed with the norm given by
Elliptic reconstruction and a posteriori error estimates for parabolic problems
 SIAM J. Numer. Anal
"... Abstract. We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, ..."
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Cited by 43 (12 self)
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Abstract. We derive a posteriori error estimates for fully discrete approximations to solutions of linear parabolic equations. The space discretization uses finite element spaces that are allowed to change in time. Our main tool is an appropriate adaptation of the elliptic reconstruction technique, introduced by Makridakis and Nochetto. We derive novel a posteriori estimates for the norms of L∞(0,T;L2(Ω)) and the higher order spaces, L∞(0,T;H1 (Ω)) and H1 (0,T;L2(Ω)), with optimal orders of convergence. 1.
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.
Numerical solution of the scalar doublewell problem allowing microstructure
 Math. Comp
, 1997
"... Abstract. The direct numerical solution of a nonconvex variational problem (P) typically faces the difficulty of the finite element approximation of rapid oscillations. Although the oscillatory discrete minimisers are properly related to corresponding Young measures and describe real physical pheno ..."
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Cited by 37 (8 self)
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Abstract. The direct numerical solution of a nonconvex variational problem (P) typically faces the difficulty of the finite element approximation of rapid oscillations. Although the oscillatory discrete minimisers are properly related to corresponding Young measures and describe real physical phenomena, they are costly and difficult to compute. In this work, we treat the scalar doublewell problem by numerical solution of the relaxed problem (RP) leading to a (degenerate) convex minimisation problem. The problem (RP) has a minimiser u and a related stress field σ = DW ∗ ∗ (∇u) which is known to coincide with the stress field obtained by solving (P) in a generalised sense involving Young measures. If uh is a finite element solution, σh: = DW ∗ ∗ (∇uh) is the related discrete stress field. We prove a priori and a posteriori estimates for σ − σh in L 4/3 (Ω) and weaker weighted estimates for ∇u −∇uh. The a posteriori estimate indicates an adaptive scheme for automatic mesh refinements as illustrated in numerical experiments. 1.
The finite element approximation of the nonlinear poissonboltzmann equation
 SIAM Journal on Numerical Analysis
"... ABSTRACT. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson–Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson–Boltzmann equation is introduced as an auxiliary problem, making it possible t ..."
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Cited by 33 (13 self)
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ABSTRACT. A widely used electrostatics model in the biomolecular modeling community, the nonlinear Poisson–Boltzmann equation, along with its finite element approximation, are analyzed in this paper. A regularized Poisson–Boltzmann equation is introduced as an auxiliary problem, making it possible to study the original nonlinear equation with delta distribution sources. A priori error estimates for the finite element approximation are obtained for the regularized Poisson–Boltzmann equation based on certain quasiuniform grids in two and three dimensions. Adaptive finite element approximation through local refinement driven by an a posteriori error estimate is shown to converge. The Poisson–Boltzmann equation does not appear to have been previously studied in detail theoretically, and it is hoped that this paper will help provide molecular modelers with a better foundation for their analytical and computational work with the Poisson–Boltzmann equation. Note that this article apparently gives the first rigorous convergence result for a numerical discretization technique for the nonlinear Poisson– Boltzmann equation with delta distribution sources, and it also introduces the first provably convergent adaptive method for the equation. This last result is currently one of only a handful of existing convergence results of this type for nonlinear problems.
A posteriori analysis of the finite element discretization of some parabolic equations
 MR2136996 RECONSTRUCTION FOR DISCRETE PARABOLIC PROBLEMS 1657
, 2005
"... Abstract. We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respe ..."
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Cited by 32 (5 self)
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Abstract. We are interested in the discretization of parabolic equations, either linear or semilinear, by an implicit Euler scheme with respect to the time variable and finite elements with respect to the space variables. The main result of this paper consists of building error indicators with respect to both time and space approximations and proving their equivalence with the error, in order to work with adaptive time steps and finite element meshes. Résumé. Nous considérons la discrétisation d’équations paraboliques, soit linéaires soit semilinéaires, par un schéma d’Euler implicite en temps et par éléments finis en espace. L’idée de cet article est de construire des indicateurs d’erreur liés à l’approximation en temps et en espace et de prouver leur équivalence avec l’erreur, dans le but de travailler avec des pas de temps adaptatifs et des maillages d’éléments finis adaptés à la solution. 1.