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Convergence of the mimetic finite difference method for diffusion problems on polyhedral meshes
 SIAM J. Numer. Anal
, 2007
"... The stability and convergence properties of the mimetic finite difference method for diffusiontype problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved. 1 ..."
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Cited by 95 (20 self)
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The stability and convergence properties of the mimetic finite difference method for diffusiontype problems on polyhedral meshes are analyzed. The optimal convergence rates for the scalar and vector variables in the mixed formulation of the problem are proved. 1
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.
An anisotropic sparse grid stochastic collocation method for elliptic partial differential equations with random input data
, 2007
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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
Identifying Vector Field Singularities Using a Discrete Hodge Decomposition
, 2002
"... this paper we use a slightly more general definition of the spaces S h respectively S # h , namely we include functions which are only defined at vertices respectively at edge midpoints. For example, the (total) Gau curvature is defined solely at vertices. Here for a given vector field # we will hav ..."
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Cited by 79 (5 self)
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this paper we use a slightly more general definition of the spaces S h respectively S # h , namely we include functions which are only defined at vertices respectively at edge midpoints. For example, the (total) Gau curvature is defined solely at vertices. Here for a given vector field # we will have div h # S h (respectively div # h # S # h ) to be defined solely at a vertex. The motivation of this generalization is twofold: first, a simplified notation of many statements, and, second, the fact that for visualization purposes one often extends these pointbased values over the surface. For example, barycentric interpolation allows to color the interior of triangles based on the discrete Gauss curvature at its vertices. Caution should be taken if integral entities are derived
Convergence of adaptive finite element methods
 SIAM Review
"... Abstract. We prove convergence of adaptive finite element methods (AFEM) for general (nonsymmetric) second order linear elliptic PDE, thereby extending the result of Morin et al [6, 7]. The proof relies on quasiorthogonality, which accounts for the bilinear form not being a scalar product, together ..."
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Cited by 72 (6 self)
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Abstract. We prove convergence of adaptive finite element methods (AFEM) for general (nonsymmetric) second order linear elliptic PDE, thereby extending the result of Morin et al [6, 7]. The proof relies on quasiorthogonality, which accounts for the bilinear form not being a scalar product, together with novel error and oscillation reduction estimates, which now do not decouple. We show that AFEM is a contraction for the sum of energy error plus oscillation. Numerical experiments, including oscillatory coefficients and convectiondiffusion PDE, illustrate the theory and yield optimal meshes.
A posteriori error estimates for nonlinear problems. Finite element discretizations of elliptic equations
 475 (1994) MR 94j:65136
"... Abstract. Using the abstract framework of [9] we analyze a residual a posteriori error estimator for spacetime finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizat ..."
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Cited by 71 (2 self)
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Abstract. Using the abstract framework of [9] we analyze a residual a posteriori error estimator for spacetime finite element discretizations of quasilinear parabolic pdes. The estimator gives global upper and local lower bounds on the error of the numerical solution. The finite element discretizations in particular cover the socalled θscheme, which includes the implicit and explicit Euler methods and the CrankNicholson scheme. 1.
Averaging techniques yield reliable a posteriori finite element error control for obstacle problems
, 2001
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Moving Least Square Reproducing Kernel Method (III): Wavelet Packet Its Applications
, 1997
"... This work is a natural extension of the work done in Part II of this series. A new partition of unity  the synchronized reproducing kernel (SRK) interpolantis proposed within the framework of moving least square reproducing kernel representation. It is a further development and generalization ..."
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Cited by 64 (13 self)
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This work is a natural extension of the work done in Part II of this series. A new partition of unity  the synchronized reproducing kernel (SRK) interpolantis proposed within the framework of moving least square reproducing kernel representation. It is a further development and generalization of the reproducing kernel particle method (RKPM), which demonstrates some superior computational capability in multiple scale numerical simulations. To form such an interpolant, a class of new wavelet functions are introduced in an unconventional way, and they form an independent sequence that is referred to as the wavelet packet. By choosing different combinations in the wavelet series expansion, the desirable synchronized convergence effect in interpolation can be achieved. Based upon the builtin consistency conditions, the differential consistency conditions for the wavelet functions are derived. It serves as an indispensable instrument in establishing the interpolation error estimate, w...
Adaptive Finite Element Methods For Optimal Control Of Partial Differential Equations: Basic Concept
 SIAM J. Contr. Optim
, 1998
"... A new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations. The Lagrangian formalism yields the firstorder necessary optimality condition in form of an indefinite boundary value problem whic ..."
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Cited by 61 (4 self)
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A new approach to error control and mesh adaptivity is described for the discretization of optimal control problems governed by elliptic partial differential equations. The Lagrangian formalism yields the firstorder necessary optimality condition in form of an indefinite boundary value problem which is approximated by an adaptive Galerkin finite element method. The mesh design in the resulting reduced models is controlled by residualbased a posteriori error estimates. These are derived by duality arguments employing the cost functional of the optimization problem for controlling the discretization error. In this case, the computed state and costate variables can be used as sensitivity factors multiplying the local cellresiduals in the error estimators. This results in a generic and simple algorithm for mesh adaptation within the optimization process. This method is developed and tested for simple boundary control problems in semiconductor models.