Partial Differential Equations - Modelling and ... - ResearchGate
Partial Differential Equations - Modelling and ... - ResearchGate Partial Differential Equations - Modelling and ... - ResearchGate
164 G. Bencteux et al. [Goe99] [HL07] [HZ05] [Koh59] [Koh96] [LeB05] Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara, C. Gonzalez, M. Challacombe, P. M. W. Gill, B. G. Johnson, W. Chen, M. W. Wong, J. L. Andres, M. Head-Gordon, E. S. Replogle, and J. A. Pople. Gaussian 98 (Revision A.7). Gaussian Inc., Pittsburgh, PA, 1998. S. Goedecker. Linear scaling electronic structure methods. Rev. Mod. Phys., 71:1085–1123, 1999. U. L. Hetmaniuk and R. B. Lehoucq. Multilevel methods for eigenspace computations in structural dynamics. In Domain Decomposition Methods in Science and Engineering XVI, volume 55 of Lect. Notes Comput. Sci. Eng., pages 103–113, Springer, Berlin, 2007. W. Hager and H. Zhang. A new conjugate gradient method with guaranteed descent and an efficient line search. SIAM J. Optim., 16:170–192, 2005. W. Kohn. Analytic properties of Bloch waves and Wannier functions. Phys. Rev., 115:809–821, 1959. W. Kohn. Density functional and density matrix method scaling linearly with the number of atoms. Phys. Rev. Lett., 76:3168–3171, 1996. C. Le Bris. Computational chemistry from the perspective of numerical analysis. In Acta Numerica, Volume 14, pages 363–444. 2005. [LNV93] X.-P. Li, R. W. Nunes, and D. Vanderbilt. Density-matrix electronic structure method with linear system size scaling. Phys. Rev. B, 47:10891– 10894, 1993. [PA04] W. P. Petersen and P. Arbenz. Introduction to Parallel Computing. Oxford University Press, 2004. [PS75] [SC00] C. Paige and M. Saunders. Solution of sparse indefinite systems of linear equations. SIAM J. Numer. Anal., 12:617–629, 1975. E. Schwegler and M. Challacombe. Linear scaling computation of the Fock matrix. Theor. Chem. Acc., 104:344–349, 2000. [YL95] W. Yang and T. Lee. A density-matrix divide-and-conquer approach for electronic structure calculations of large molecules. J. Chem. Phys., 163:5674, 1995.
Numerical Analysis of a Finite Element/Volume Penalty Method Bertrand Maury Laboratoire de Mathématiques, Université Paris-Sud, FR-91405 Orsay Cedex, France Bertrand.Maury@math.u-psud.fr Summary. The penalty method makes it possible to incorporate a large class of constraints in general purpose Finite Element solvers like freeFEM++. We present here some contributions to the numerical analysis of this method. We propose an abstract framework for this approach, together with some general error estimates based on the discretization parameter ε and the space discretization parameter h. As this work is motivated by the possibility to handle constraints like rigid motion for fluid-particle flows, we shall pay a special attention to a model problem of this kind, where the constraint is prescribed over a subdomain. We show how the abstract estimate can be applied to this situation, in the case where a non-body-fitted mesh is used. In addition, we describe how this method provides an approximation of the Lagrange multiplier associated to the constraint. 1 Introduction Because of its conceptual simplicity and the fact that it is usually straightforward to implement, the penalty method has been widely used to incorporate constraints in numerical optimization. The general principle can been seen as a relaxed version of the following fact: given a proper functional J over a set X, andK a subset of X, minimizing J over K is equivalent to minimizing J K = J + I K over X, whereI K is the indicatrix of K: { 0 if x ∈ K I K (x) = +∞ if x/∈ K Assume now that K can be defined as K = {x ∈ X | Ψ(x) =0}, whereΨ is a non-negative function, the penalty method consists in considering relaxed functionals J ε defined as J ε = J + 1 Ψ, ε > 0. ε By definition of K, the function Ψ/ε approaches I K point-wise:
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164 G. Bencteux et al.<br />
[Goe99]<br />
[HL07]<br />
[HZ05]<br />
[Koh59]<br />
[Koh96]<br />
[LeB05]<br />
Martin, D. J. Fox, T. Keith, M. A. Al-Laham, C. Y. Peng, A. Nanayakkara,<br />
C. Gonzalez, M. Challacombe, P. M. W. Gill, B. G. Johnson, W. Chen,<br />
M. W. Wong, J. L. Andres, M. Head-Gordon, E. S. Replogle, <strong>and</strong> J. A. Pople.<br />
Gaussian 98 (Revision A.7). Gaussian Inc., Pittsburgh, PA, 1998.<br />
S. Goedecker. Linear scaling electronic structure methods. Rev. Mod.<br />
Phys., 71:1085–1123, 1999.<br />
U. L. Hetmaniuk <strong>and</strong> R. B. Lehoucq. Multilevel methods for eigenspace<br />
computations in structural dynamics. In Domain Decomposition Methods<br />
in Science <strong>and</strong> Engineering XVI, volume 55 of Lect. Notes Comput. Sci.<br />
Eng., pages 103–113, Springer, Berlin, 2007.<br />
W. Hager <strong>and</strong> H. Zhang. A new conjugate gradient method with guaranteed<br />
descent <strong>and</strong> an efficient line search. SIAM J. Optim., 16:170–192,<br />
2005.<br />
W. Kohn. Analytic properties of Bloch waves <strong>and</strong> Wannier functions.<br />
Phys. Rev., 115:809–821, 1959.<br />
W. Kohn. Density functional <strong>and</strong> density matrix method scaling linearly<br />
with the number of atoms. Phys. Rev. Lett., 76:3168–3171, 1996.<br />
C. Le Bris. Computational chemistry from the perspective of numerical<br />
analysis. In Acta Numerica, Volume 14, pages 363–444. 2005.<br />
[LNV93] X.-P. Li, R. W. Nunes, <strong>and</strong> D. V<strong>and</strong>erbilt. Density-matrix electronic<br />
structure method with linear system size scaling. Phys. Rev. B, 47:10891–<br />
10894, 1993.<br />
[PA04] W. P. Petersen <strong>and</strong> P. Arbenz. Introduction to Parallel Computing.<br />
Oxford University Press, 2004.<br />
[PS75]<br />
[SC00]<br />
C. Paige <strong>and</strong> M. Saunders. Solution of sparse indefinite systems of linear<br />
equations. SIAM J. Numer. Anal., 12:617–629, 1975.<br />
E. Schwegler <strong>and</strong> M. Challacombe. Linear scaling computation of the<br />
Fock matrix. Theor. Chem. Acc., 104:344–349, 2000.<br />
[YL95] W. Yang <strong>and</strong> T. Lee. A density-matrix divide-<strong>and</strong>-conquer approach<br />
for electronic structure calculations of large molecules. J. Chem. Phys.,<br />
163:5674, 1995.