Abstract: In this paper, we study the stabilizer-free weak Galerkin methods on polytopal meshes for a class of second order elliptic boundary value problems of divergence form and with gradient nonlinearity in the principal coefficient. With certain assumptions on the nonlinear coefficient, we show that the discrete problem has a unique solution. This is achieved by showing that the associated operator satisfies certain continuity and monotonicity properties. With the help of these properties, we derive optimal error estimates in the energy norm. We present several numerical examples to verify the error estimates.
@article{Ye.X;Zhang.S;Zhu.Y2020,
Author = {Xiu Ye, Shangyou Zhang and Yunrong Zhu},
Doi = {10.1016/j.rinam.2020.100097},
Journal = {Results in Applied Mathematics},
Title = {Stabilizer-free weak Galerkin methods for monotone
quasilinear elliptic PDEs},
Year = {2020},
}
Abstract: We present a linear algebra approach to establishing a discrete comparison principle for a nonmonotone class of quasilinear elliptic partial differential equations. In the absence of a lower order term, local conditions on the mesh are required to establish the comparison principle and uniqueness of the piecewise linear finite element solution. We consider the assembled matrix corresponding to the linearized problem satisfied by the difference of two solutions to the nonlinear problem. Monotonicity of the assembled matrix establishes a maximum principle for the linear problem and a comparison principle for the nonlinear problem. The matrix analysis approach to the discrete comparison principle yields sharper constants and more relaxed mesh conditions than does the argument by contradiction used in previous work.
@article{Pollock.S;Zhu.Y2019,
Author = {Sara Pollock and Yunrong Zhu},
Doi = {10.1007/s11075-019-00713-x},
Journal = {Numerical Algorithms},
Title = {A matrix analysis approach to discrete comparison principles
for nonmonotone PDE},
Year = {2019},
}
Abstract: Uniqueness of the finite element solution for nonmonotone quasilinear problems of elliptic type is established in one and two dimensions. In each case, we prove a comparison theorem based on locally bounding the variation of the discrete solution over each element. The uniqueness follows from this result, and does not require a globally small meshsize.
@article{Pollock.S;Zhu.Y2018,
Author = {Sara Pollock and Yunrong Zhu},
Doi = {10.1007/s00211-018-0956-4},
Journal = {Numer. Math.},
Title = {Uniqueness of discrete solutions of nonmonotone PDEs
without a globally fine mesh condition},
Year = {2018},
}
Abstract: Arnold, Falk, and Winther [Bull. Amer. Math. Soc. 47 (2010), 281–354] showed that
mixed variational problems, and their numerical approximation by mixed methods, could
be most completely understood using the ideas and tools of Hilbert complexes. This led to
the development of the Finite Element Exterior Calculus (FEEC) for a large class of linear
elliptic problems. More recently, Holst and Stern [Found. Comp. Math. 12:3 (2012), 263–
293 and 363–387] extended the FEEC framework to semi-linear problems, and to problems
containing variational crimes, allowing for the analysis and numerical approximation of
linear and nonlinear geometric elliptic partial differential equations on Riemannian manifolds
of arbitrary spatial dimension, generalizing surface finite element approximation
theory. In this article, we develop another distinct extension to the FEEC, namely to
parabolic and hyperbolic evolution systems, allowing for the treatment of geometric and
other evolution problems. Our approach is to combine the recent work on the FEEC for
elliptic problems with a classical approach to solving evolution problems via semi-discrete
finite element methods, by viewing solutions to the evolution problem as lying in timeparameterized
Hilbert spaces (or Bochner spaces). Building on classical approaches by
Thom´ee for parabolic problems and Geveci for hyperbolic problems, we establish a priori
error estimates for Galerkin FEM approximation in the natural parametrized Hilbert space
norms. In particular, we recover the results of Thom´ee and Geveci for two-dimensional
domains and lowest-order mixed methods as special cases, effectively extending their results
to arbitrary spatial dimension and to an entire family of mixed methods. We also
show how the Holst and Stern framework allows for extensions of these results to certain
semi-linear evolution problems.
@article{Gillette.A;Holst.M;Zhu.Y2017,
Author = {Andrew Gillette and Michael Holst and Yunrong Zhu},
Doi = {10.4208/jcm.1610-m2015-0319},
Journal = {J. Comp. Math.},
Number = {2},
Pages = {187-212},
Title = {Finite Element Exterior Calculus for Evolution Problems},
Volume = {35},
Year = {2017},
}
Abstract: In this paper, we extend some of the multilevel convergence results obtained by Xu
and Zhu in [Xu and Zhu, M3AS 2008], to the case of second order linear reaction-diffusion
equations. Specifically, we consider the multilevel preconditioners for solving the linear
systems arising from the linear finite element approximation of the problem, where both
diffusion and reaction coefficients are piecewise-constant functions. We discuss in detail the
influence of both the discontinuous reaction and diffusion coefficients to the performance of
the classical BPX and multigrid V-cycle preconditioner.
@article{Kolev.T;Xu.J;Zhu.Y2016,
Author = {Kolev, Tzanio V. and Xu, Jinchao and Zhu, Yunrong},
Doi = {10.1007/s10915-015-0083-7},
Journal = {Journal of Scientific Computing},
Month = {Apr},
Number = {1},
Pages = {324--350},
Title = {Multilevel Preconditioners for Reaction-Diffusion Problems
with Discontinuous Coefficients},
Volume = {67},
Year = {2016},
}
Abstract: In this article we develop a convergence theory for goal-oriented adaptive finite element algorithms designed for a class of second-order semilinear elliptic equations. We briefly discuss the target problem class, and introduce several related approximate dual problems that are crucial to both the analysis as well as to the development of a practical numerical method. We then review some standard facts concerning conforming finite element discretization and error-estimate-driven adaptive finite element methods (AFEM). We include a brief summary of a priori estimates for this class of semilinear problems, and then describe some goal-oriented variations of the standard approach to AFEM. Following the recent approach of Mommer--Stevenson and Holst--Pollock for increasingly general linear problems, we first establish a quasi-error contraction result for the primal problem. We then develop some additional estimates that make it possible to establish contraction of the combined primal-dual quasi-error, and subsequently show convergence with respect to the quantity of interest. Finally, a sequence of numerical experiments are examined and it is observed that the behavior of the implementation follows the predictions of the theory.
@article{Holst.M;Pollock.S;Zhu.Y2015,
Author = {Holst, Michael and Pollock, Sara and Zhu, Yunrong},
Doi = {10.1007/s00791-015-0243-1},
Journal = {Computing and Visualization in Science},
Number = {1},
Pages = {43--63},
Title = {Convergence of goal-oriented adaptive finite element
methods for semilinear problems},
Volume = {17},
Year = {2015},}
Abstract: The goal of this paper is to design
optimal multilevel solvers for the finite element approximation
of second order linear elliptic problems with piecewise constant
coefficients on bisection grids. Local multigrid and BPX preconditioners
are constructed based on local smoothing only at the newest vertices
and their immediate neighbors. The analysis of eigenvalue distributions
for these local multilevel preconditioned systems shows that there are
only a fixed number of eigenvalues which are deteriorated by the large
jump. The remaining eigenvalues are bounded uniformly with respect to
the coefficients and the meshsize. Therefore, the resulting
preconditioned conjugate gradient algorithm will converge with an
asymptotic rate independent of the coefficients and logarithmically
with respect to the meshsize. As a result, the overall computational
complexity is nearly optimal.
@article{Chen.L;Holst.M;Xu.J;Zhu.Y2013,
Author = {Chen, L., Holst, M., Xu, J. and Zhu, Y.},
Journal = {Computing and Visualization in Science
(Accepted)},
Title = {Local Multilevel Preconditioners for Elliptic Equations
with Jump Coefficients on Bisection Grids.},
Year = {2012},
DOI = {10.1007/s00791-013-0213-4}
}
Abstract: In this article we consider two-grid
finite element methods for solving semilinear interface problems
in d space dimensions, for d = 2 or d = 3. We
consider semilinear problems with discontinuous diffusion coefficients,
which includes problems containing sub-critical, critical, and
supercritical nonlinearities. We establish basic quasi-optimal a
priori error estimates for Galerkin approximations. We then design a
two-grid algorithm consisting of a coarse grid solver for the original
nonlinear problem, and a fine grid solver for a linearized problem. We
analyze the quality of approximations generated by the algorithm, and
show that the coarse grid may be taken to have much larger elements
than the fine grid, and yet one can still obtain approximation quality
that is asymptotically as good as solving the original nonlinear
problem on the fine mesh.
@article{Holst.M;Szypowski.R;Zhu.Y2012,
Author = {Holst, M., Szypowski, R. and Zhu, Y.},
Journal = {Numerical Methods for Partial Differential Equations
(Accepted)},
Title = {Two-grid Methods for Semilinear Interface Problems},
Year = {2012},
}
Abstract: In this article we study adaptive finite
element methods (AFEM) with inexact solvers for a class of semilinear
elliptic interface problems. We are particularly interested in nonlinear
problems with discontinuous diffusion coefficients, such as the nonlinear
Poisson-Boltzmann equation and its regularizations. The algorithm we
study consists of the standard SOLVE-ESTIMATE-MARK-REFINE procedure
common to many adaptive finite element algorithms, but where the SOLVE
step involves only a full solve on the coarsest level, and the
remaining levels involve only single Newton updates to the previous
approximate solution. We summarize a recently developed AFEM
convergence theory for inexact solvers, and present a sequence of
numerical experiments that give evidence that the theory does in
fact predict the contraction properties of AFEM with inexact solvers.
The various routines used are all designed to maintain a linear-time
computational complexity.
@article{Holst.M;Szypowski.R;Zhu.Y2012,
Author = {Holst, M., Szypowski, R. and Zhu, Y.},
Journal = {Domain Decomposition Methods in Science and Engineering XX},
Title = {Adaptive Finite Element Methods with Inexact Solvers for the
Nonlinear {Poisson-Boltzmann} Equation},
Year = {2012},
}
Abstract: In this paper, we present a multigrid
preconditioner for solving the linear system arising from the
piecewise linear nonconforming Crouzeix- Raviart discretization of
second order elliptic problems with jump coefficients. The
preconditioner uses the standard conforming subspaces as coarse spaces.
Numerical tests show both robustness with respect to the jump in the
coefficient and near-optimality with respect to the number of
degrees of freedom.
@article{Ayuso-de-Dios.B;Holst.M;Zhu.Y;Zikatanov.L2012a,
Author = {Ayuso de Dios, B. and Holst, M. and Zhu, Y.
and Zikatanov, L.},
Journal = {Domain Decomposition Methods in Science and Engineering XX},
Title = {Multigrid Preconditioner for Nonconforming Discretization
of Elliptic Problems with Jump Coefficients},
Year = {2012},
}
Abstract: We introduce and analyze two-level and
multi-level preconditioners for a family of Interior Penalty (IP)
discontinuous Galerkin (DG) discretizations of second order elliptic
problems with large jumps in the diffusion coefficient. Our approach
to IPDG-type methods is based on a splitting of the DG space into two
components that are orthogonal in the energy inner product naturally
induced by the methods. As a result, the methods and their analysis
depend in a crucial way on the diffusion coefficient of the problem.
The analysis of the pro- posed preconditioners is presented for both
symmetric and non-symmetric IP schemes; dealing simultaneously with
the jump in the diffusion coefficient and the non-nested character
of the relevant discrete spaces presents additional difficulties in
the analysis, which precludes a simple extension of existing results.
However, we are able to establish robustness (with respect to the
diffusion coefficient) and near-optimality (up to a logarithmic term
depending on the mesh size) for both two-level and BPX-type
preconditioners, by using a more refined Conjugate Gradi- ent theory.
Useful by-products of the analysis are the supporting results on the
construction and analysis of simple, efficient and robust two-level
and multilevel preconditioners for noncon- forming Crouzeix-Raviart
discretizations of elliptic problems with jump coefficients.
Following the analysis, we present a sequence of detailed numerical
results which verify the theory and illustrate the performance of
the methods.
@article{Ayuso-de-Dios.B;Holst.M;Zhu.Y;Zikatanov.L2012,
Author = {Ayuso de Dios, B. and Holst, M. and Zhu, Y.
and Zikatanov, L.},
Journal = {Mathematics Computation (In press)},
Title = {{Multilevel preconditioners for discontinuous Galerkin
approximations of elliptic problems with jump coefficients}},
Year = {2012},
}
Abstract:
In this paper, we present a multigrid V-cycle preconditioner for
the linear system arising from piecewise linear nonconforming
Crouzeix–Raviart discretization of second-order elliptic problems
with jump coefficients. The preconditioner uses standard conforming
subspaces as coarse spaces. We showed that the convergence rates of
the (multiplicative) two-grid and multigrid V-cycle algorithms will
deteriorate rapidly because of large jumps in coefficient. However,
the preconditioned systems have only a fixed number of small eigenvalues
depending on the large jump in coefficient, and the effective condition
numbers are independent of the coefficient and bounded logarithmically
with respect to the mesh size. As a result, the two-grid or multigrid
preconditioned conjugate gradient algorithm converges nearly uniformly.
We also comment on some major differences of the convergence theory
between the nonconforming case and the standard conforming case.
Numerical experiments support the theoretical results.
@article{Zhu.Y2012,
Author = {Y. Zhu},
Journal = {Numerical Linear Algebra with Applications},
DOI ={DOI: 10.1002/nla.1856},
Title = {Analysis of a multigrid preconditioner for Crouzeix–Raviart
discretization of elliptic partial differential equation with jump
coefficients},
Year = {2012}}
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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Contact
Yunrong Zhu Email:zhuyunr@isu.edu Phone: (208)282-3819(O) Office: Physical Sciences 328B
Department of Mathematics & Statistics
Idaho State University
921 South 8th Avenue, P.O. Box 8085
Pocatello, ID 83209, USA