Abstract: We consider the design of an effective and
reliable adaptive finite element method (AFEM) for the nonlinear
Poisson-Boltzmann equation (PBE). We first examine the two-term
regularization technique for the continuous problem recently proposed
by Chen, Holst, and Xu based on the removal of the singular
electrostatic potential inside biomolecules; this technique made
possible the development of the first complete solution and
approximation theory for the Poisson-Boltzmann equation, the first
provably convergent discretization, and also allowed for the
development of a provably convergent AFEM. However, in practical
implementation, this two-term regularization exhibits numerical
instability. Therefore, we examine a variation of this regularization
technique which can be shown to be less susceptible to such instability.
We establish a priori estimates and other basic results for the
continuous regularized problem, as well as for Galerkin finite element
approximations. We show that the new approach produces regularized
continuous and discrete problems with the same mathematical advantages
of the original regularization. We then design an AFEM scheme for the
new regularized problem, and show that the resulting AFEM scheme is
accurate and reliable, by proving a contraction result for the error.
This result, which is one of the first results of this type for
nonlinear elliptic problems, is based on using continuous and discrete
a priori L∞ estimates to establish quasi- orthogonality. To provide a
high-quality geometric model as input to the AFEM algorithm, we also
describe a class of feature-preserving adaptive mesh generation
algorithms designed specifically for constructing meshes of biomolecular
structures, based on the intrinsic local structure tensor of the molecular
surface. All of the algorithms described in the article are implemented
in the Finite Element Toolkit (FETK), developed and maintained at UCSD.
The stability advantages of the new regularization scheme are
demonstrated with FETK through comparisons with the original
regularization approach for a model problem. The convergence and
accuracy of the overall AFEM algorithm is also illustrated by numerical
approximation of electrostatic solvation energy for an insulin protein.

@article{Holst.M;McCammon.J;Yu.Z;Zhou.Y2009,
Author = {M. Holst and J.A. McCammon and Z. Yu and Y.C. Zhou
and Y. Zhu},
Journal = {Communications in Computational Physics},
DOI = {doi:10.4208/cicp.081009.130611a},
Title = {{Adaptive Finite Element Modeling Techniques for
the Poisson-Boltzmann Equation}},
Volumne = {11},
Number = {1},
Pages = {179-214},
Year = {2012},
}

Abstract: In this paper, we construct an auxiliary
space preconditioner for Maxwell‚Äôs equations with interface, and
generalize the HX preconditioner developed in [9] to the problem with
strongly discontinuous coefficients. For the H ( curl ) interface
problem, we show that the condition number of the HX preconditioned
system is uniformly bounded with respect to the coefficients and
meshsize.

@incollection{Xu.J;Zhu.Y2011,
Author = {Xu, J. and Zhu, Y.},
Booktitle = {Domain Decomposition Methods in Science
and Engineering XIX},
Editor = {Huang, Yunqing and Kornhuber, Ralf
and Widlund, Olof and Xu, Jinchao},
Pages = {173-180},
Publisher = {Springer Berlin Heidelberg},
Series = {Lecture Notes in Computational Science and Engineering},
Title = {Robust Preconditioner for H(curl) Interface Problems},
Volume = {78},
Year = {2011}}

Abstract: This paper is devoted to study of an
auxiliary spaces preconditioner for H(div) systems and its application
in the mixed formulation of second order elliptic equations. Extensive
numerical results show the efficiency and robustness of the algorithms,
even in the presence of large coefficient variations. For the mixed
formulation of elliptic equations, we use the augmented Lagrange
technique to convert the solution of the saddle point problem into the
solution of a nearly singular H(div) system. Numerical experiments also
justify the robustness and efficiency of this scheme.

@incollection {Tuminaro.R;Xu.J;Zhu.Y2009,
author = {R. Tuminaro and J. Xu and Y. Zhu},
title = {Auxiliary Space Preconditioners for Mixed
Finite Element Methods},
booktitle = {Domain Decomposition Methods in Science
and Engineering XVIII},
series = {Lecture Notes in Computational Science and Engineering},
publisher = {Springer Berlin Heidelberg},
pages = {99-109},
volume = {70},
year = {2009}
}

Abstract: This paper gives a solution to an open problem
concerning the performance of various multilevel preconditioners for the
linear finite element approximation of second-order elliptic boundary
value problems with strongly discontinuous coefficients. By analyzing
the eigenvalue distribution of the BPX preconditioner and multigrid
V-cycle precondi- tioner, we prove that only a small number of
eigenvalues may deteriorate with respect to the discontinuous jump
or meshsize, and we prove that all the other eigenvalues are bounded
below and above nearly uniformly with respect to the jump and meshsize.
As a result, we prove that the convergence rate of the preconditioned
conjugate gradient methods is uniform with respect to the large jump
and meshsize. We also present some numerical experiments to demonstrate
the theoretical results.

@article{Xu.J;Zhu.Y2008,
Author = {Xu, J. and Zhu, Y.},
Journal = {Mathematical Models and Methods in Applied Science},
Number = {1},
Pages = {77 --105},
Title = {Uniform convergent multigrid methods for elliptic
problems with strongly discontinuous coefficients},
Volume = {18},
Year = {2008}}

Abstract:
This paper provides a proof of the robustness of the
overlapping domain decomposition preconditioners for the linear finite
element approximation of second order elliptic boundary value problems
with strongly discontinuous coefficients. By analyzing the eigenvalue
distribution of the domain decomposition preconditioner, we prove that
only a small number of eigenvalues may deteriorate with respect to the
discontinuous jump or meshsize, and all the other eigenvalues are
bounded below and above nearly uniformly with respect to the jump
and meshsize. As a result, we prove that the asymptotic convergence
rate of the preconditioned conjugate gradient methods is uniform with
respect to the large jump and meshsize.

@article{Zhu.Y2008,
Author = {Y. Zhu},
Journal = {Numerical Linear Algebra with Applications},
Number = {2-3},
Pages = {271-289},
Title = {Domain Decomposition Preconditioners for Elliptic
Equations with Jump Coefficients},
Volume = {15},
Year = {2008}}

Abstract:
We are concerned with the compatible gauge reformulation for H(div)
equations and the design of fast solvers of the resulting linear
algebraic systems as in [5]. We propose an algebraic reformulation
of the discrete H(div) equations along with an algebraic multigrid
(AMG) technique for the reformulated problem. The reformulation uses
discrete Hodge decompositions on co-chains to replace the discrete
H(div) equations by an equivalent 2×2 block linear system whose
diagonal blocks are discrete Hodge Laplace operators acting on
2-cochains and 1-cochains respectively. We illustrate the new
technique, using the lowest order Raviart-Thomas elements on
structured tetrahedral mesh in three dimension, and present
compuutational results.

@inproceedings{Bochev.P;Seifert.C;Tuminaro.R;Xu.J2007,
Author = {P. Bochev and C. Siefert and R. Tuminaro
and J. Xu and Y. Zhu},
Booktitle = {CSRI Summer Proceedings},
Title = {Compatible gauge approaches for {$H({\rm div})$}
equations},
Year = {2007}}

Abstract:
This dissertation is devoted to practical design and theoretical
analysis of efficient and robust preconditioners for solving
algebraic systems arising from the approximation of partial
differential equations, with special emphasis on the problems
with strongly discontinuous coefficients. The problems considered
here include the standard second order elliptic equations (H(grad)
or H1 equations), as well as the second order elliptic systems given
in terms of curl and divergence operators (H(curl) and H(div) systems).

@phdthesis{Zhu.Y2008a,
Author = {Yunrong Zhu},
Month = {August},
School = {The Pennsylvania State University},
Title = {Robust preconditioners for H(grad), H(curl)
and H(div) systems with strongly discontinuous coefficients},
Year = {2008}}

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Yunrong Zhu Email:zhu@math.ucsd.edu Phone: (858)534-5862(O) Office: AP&M 5824
Department of Mathematics
University of California, San Diego
9500 Gilman Drive, Dept. # 0112
La Jolla, CA 92093, USA