Learn More
No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this(More)
The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonal-ity. A quasi-orthogonality property is proved using the fact that the error is orthogonal to the(More)
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 mani-folds is first reviewed, and then adaptive multilevel finite element methods for approximating solutions to this class of problems are(More)
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(More)
A multigrid method is presented for the numerical solution of the linearized Poisson-Boltzmann equation arising in molecular biophysics. The equation is discretized with the finite volume method, and the numerical solution of the discrete equations is accomplished with multiple grid techniques originally developed for two-dimensional interface problems(More)
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(More)
This paper is the first of two papers on the adaptive multilevel finite element treatment of the nonlinear Poisson-Boltzmann equation (PBE), a nonlinear elliptic equation arising in biomolecular modeling. Fast and accurate numerical solution of the PBE is usually difficult to accomplish, due to presence of discontinuous coefficients, delta functions, three(More)
We apply the adaptive multilevel finite element techniques described in [20] to the nonlinear Poisson-Boltzmann equation (PBE) in the context of biomolecules. Fast and accurate numerical solution of the PBE in this setting is usually difficult to accomplish due to presence of discontinuous coefficients, delta functions, three spatial dimensions, unbounded(More)
We present a robust and efficient numerical method for solution of the nonlinear Poisson-Boltzmann equation arising in molecular biophysics. The equation is discretized with the box method, and solution of the discrete equations is accomplished with a global inexact-Newton method, combined with linear multilevel techniques we have described in a paper(More)
  • Bernardo Cockburn, Johnny Guzman, Haiying Wang, Long Chen, Michael Holst, Jinchao Xu +18 others
  • 2010
1 Superconvergent discontinuous Galerkin methods for second-order elliptic problems / A multiscale finite element method for partial differential equations posed in domains with rough boundaries / Alexandre L. Madureira 35 Convergence and optimality of adaptive mixed finite element methods / 79 Overlapping additive Schwarz preconditioners for elliptic PDEs(More)