Y. Zhu's Publications

Paper Published/Accepted

1. M. Holst, J. A. McCammon Z. Yu, Y.C. Zhou and Y. Zhu. Adaptive Finite Element Modeling Techniques for the Poisson-Boltzmann Equation. Communications in Computational Physics. 11 (1), 179-214 (2012) Published Online: September 8, 2011

   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},
   }
   

2. J. Xu and Y. Zhu. Robust Preconditioner for H(curl) Interface Problems. Domain Decomposition Methods in Science and Engineering XIX. 78, 173-180 (2011)

   Abstract: In this paper, we construct an auxiliary space preconditioner for Maxwell’s 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}}
   
   

3. R. Tuminaro, J. Xu and Y. Zhu. Auxiliary Space Preconditioners for Mixed Finite Element Methods. Domain Decomposition Methods in Science and Engineering XVIII. 70, 99-109 (2009)

   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}
   }
   

4. J. Xu and Y. Zhu. Uniform Convergent Multigrid Methods for Elliptic Problems with Strongly Discontinuous Coefficients. Mathematical models and Methods in Applied Science.   18(1), 77-105 (2008)

   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}}
   

5. Y. Zhu. Domain Decomposition Preconditioners for Elliptic Equations with Jump Coefficients. Numerical Linear Algebra with Applications.   15, 271-289 (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}}
   

6. P. Bochev, C. Siefert, R. Tuminaro, J. Xu and Y. Zhu, Compatible Gauge Approaches for H(div) Equations CSRI Summer Proceedings.  3-13 (2007)

   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}}
   
   

7. Y. Zhu, Robust preconditioners for H(grad), H(curl) and H(div) systems with strongly discontinuous coefficients Ph. D. Thesis. (2008)

   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}}