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42
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 43 (14 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 41 (24 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 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 25 (7 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
A Review of A Posteriori Error Estimation
 and Adaptive MeshRefinement Techniques, Wiley & Teubner
, 1996
"... linear parabolic equations ..."
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 20 (3 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.
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 17 (3 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.
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 16 (11 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 Error Estimation For The Finite Element MethodOfLines Solution Of Parabolic Problems
 Math. Models and Meths. in Appl. Sci
"... Babuska and Yu constructed a posteriori estimates for finite element discretization errors of linear elliptic problems utilizing a dichotomy principal stating that the errors of oddorder approximations arise near element edges as mesh spacing decreases while those of evenorder approximations arise ..."
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Cited by 16 (5 self)
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Babuska and Yu constructed a posteriori estimates for finite element discretization errors of linear elliptic problems utilizing a dichotomy principal stating that the errors of oddorder approximations arise near element edges as mesh spacing decreases while those of evenorder approximations arise in element interiors. We construct similar a posteriori estimates for the spatial errors of finite element methodoflines solutions of linear parabolic partial differential equations on squareelement meshes. Error estimates computed in this manner are proven to be asymptotically correct; thus, they converge in strain energy under mesh refinement at the same rate as the actual errors. 1. Introduction A posteriori estimates of discretization errors have been an integral part of adaptive finite element methods since their inception nearly twenty years ago and are used for elliptic 4;5;6;7;11;16;29;30 and parabolic problems. 3;14;15;17;20;21;24 Local contributions to global error estimat...
An adaptive finite element algorithm with reliable and efficient error control for linear parabolic problems
 Math. Comp
"... An ecient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. An adaptive algorithm with variable timestep sizes and space meshes is proposed and studied which, at each time step, ..."
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Cited by 15 (1 self)
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An ecient and reliable a posteriori error estimate is derived for linear parabolic equations which does not depend on any regularity assumption on the underlying elliptic operator. An adaptive algorithm with variable timestep sizes and space meshes is proposed and studied which, at each time step, delays the mesh coarsening until the nal iteration of the adaptive procedure, allowing only mesh and timestep size re nements before. It is proved that at each time step, the adaptive algorithm is able to reduce the error indicators (and thus the error) below any given tolerance within nite number of iteration steps. The key ingredient in the analysis is a new coarsening strategy. Numerical results are presented to show the competitive behavior of the proposed adaptive algorithm. 1.