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A Lagrange Multiplier Based Domain Decomposition Method 145 Fig. 11. Contour plot of the real part of the solution for L =0.5 (left) and L =0.25 (right). Incident wave coming from the left lower corner with a 45 degrees angle. References [BDG + 97] [DL92] [GL89] [Glo03] M. O. Bristeau, E. J. Dean, R. Glowinski, V. Kwok, and J. Périaux. Exact controllability and domain decomposition methods with nonmatching grids for the computation of scattering waves. In R. Glowinski, J. Périaux, and Z. Shi, editors, Domain Decomposition Methods in Sciences and Engineering, pages 291–307. John Wiley & Sons, 1997. R. Dautray and J.-L. Lions. Mathematical Analysis and Numerical Methods for Science and Technology, volume 5. Springer-Verlag, 1992. R. Glowinski and P. LeTallec. Augmented Lagrangian and Operator Splitting Methods in Nonlinear Mechanics. SIAM, Philadelphia, PA, 1989. R. Glowinski. Finite element methods for incompressible viscous flow. In P. G. Ciarlet and J.-L. Lions, editors, Handbook of Numerical Analysis, Vol. IX, pages 3–1176. North-Holland, Amsterdam, 2003.
Domain Decomposition and Electronic Structure Computations: A Promising Approach Guy Bencteux 1,4 , Maxime Barrault 1 , Eric Cancès 2,4 , William W. Hager 3 , and Claude Le Bris 2,4 1 EDF R&D, 1 avenue du Général de Gaulle, 92141 Clamart Cedex, France {guy.bencteux,maxime.barrault}@edf.fr 2 CERMICS, École Nationale des Ponts et Chaussées, 6 & 8, avenue Blaise Pascal, Cité Descartes, 77455 Marne-La-Vallée Cedex 2, France, {cances,lebris}@cermics.enpc.fr 3 Department of Mathematics, University of Florida, Gainesville, FL 32611-8105, USA, hager@math.ufl.edu 4 INRIA Rocquencourt, MICMAC project, Domaine de Voluceau, B.P. 105, 78153 Le Chesnay Cedex, France Summary. We describe a domain decomposition approach applied to the specific context of electronic structure calculations. The approach has been introduced in [BCHL07]. We survey here the computational context, and explain the peculiarities of the approach as compared to problems of seemingly the same type in other engineering sciences. Improvements of the original approach presented in [BCHL07], including algorithmic refinements and effective parallel implementation, are included here. Test cases supporting the interest of the method are also reported. It is our pleasure and an honor to dedicate this contribution to Olivier Pironneau, on the occasion of his sixtieth birthday. With admiration, respect and friendship. 1 Introduction and Motivation 1.1 General Context Numerical simulation is nowadays an ubiquitous tool in materials science, chemistry and biology. Design of new materials, irradiation induced damage, drug design, protein folding are instances of applications of numerical simulation. For convenience we now briefly present the context of the specific computational problem under consideration in the present article. A more detailed, mathematically-oriented, presentation is the purpose of the monograph [CDK + 03] or of the review article [LeB05]. For many problems of major interest, empirical models where atoms are represented as point particles interacting with a parameterized force-field are
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Partial Differential Equations
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Partial Differential Equations Mode
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Dedicated to Olivier Pironneau
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VIII Preface computers has been at
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Contents List of Contributors .....
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List of Contributors Yves Achdou UF
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List of Contributors XV Claude Le B
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Discontinuous Galerkin Methods Vive
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Discontinuous Galerkin Methods 5 2
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Discontinuous Galerkin Methods 7 g
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Discontinuous Galerkin Methods 9 (
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Discontinuous Galerkin Methods 15 W
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Let a h and b h denote the bilinear
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Discontinuous Galerkin Methods 19 t
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Discontinuous Galerkin Methods 21 l
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Table 1. Primal DG for transport Di
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Discontinuous Galerkin Methods 25 [
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Mixed Finite Element Methods on Pol
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Mixed FE Methods on Polyhedral Mesh
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4 Hybridization and Condensation Mi
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is symmetric and positive definite,
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with some coefficient α ∈ R wher
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44 E.J. Dean and R. Glowinski so fa
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46 E.J. Dean and R. Glowinski 2 A L
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48 E.J. Dean and R. Glowinski S:T=
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50 E.J. Dean and R. Glowinski minim
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52 E.J. Dean and R. Glowinski 6 On
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54 E.J. Dean and R. Glowinski Fig.
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56 E.J. Dean and R. Glowinski and
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58 E.J. Dean and R. Glowinski 7 Num
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60 E.J. Dean and R. Glowinski Fig.
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62 E.J. Dean and R. Glowinski Assum
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Higher Order Time Stepping for Seco
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u n+1 h Optimal Higher Order Time D
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Optimal Higher Order Time Discretiz
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200 A. Bonito et al. The normal vec
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204 A. Bonito et al. Fig. 8. Jet bu
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206 A. Bonito et al. References [AM
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208 A. Bonito et al. [Set96] J. A.
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210 J. Hao et al. due to shear flow
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212 J. Hao et al. The backward reac
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214 J. Hao et al. where u and p den
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216 J. Hao et al. * * * * * * * * *
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220 J. Hao et al. Table 2. The calc
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222 J. Hao et al. References [ASS80
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Computing the Eigenvalues of the La
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Eigenvalues of the Laplace-Beltrami
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A Fixed Domain Approach in Shape Op
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Shape Optimization Problems with Ne
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Reduced-Order Modelling of Dispersi
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Calibration of Lévy Processes with
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We have proved Calibration of Lévy
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280 S. Ikonen and J. Toivanen the p
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