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A Domain Decomposition Method for the Helmholtz equation and related Optimal Control Problems
 in Mathematical and Numerical Aspects of Wave Propagation Phenomena
, 1996
"... : We present an iterative domain decomposition method to solve the Helmholtz equation and related optimal control problems. The proof of convergence of this method relies on energy techniques. This method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problem ..."
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Cited by 101 (1 self)
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: We present an iterative domain decomposition method to solve the Helmholtz equation and related optimal control problems. The proof of convergence of this method relies on energy techniques. This method leads to efficient algorithms for the numerical resolution of harmonic wave propagation problems in heterogeneous media and their control. Keywords : Domain decomposition, Helmholtz equation, Harmonic waves, Optimal control, Waveguide, Absorbing boundary conditions AMS(MOS) subject classification : 35J05, 49M99, 65N55 (R'esum'e : tsvp) INRIA, Domaine de Voluceau, B.P.105 78153 Le Chesnay cedex, France, email :benamou @misstick.inria.fr Commissariat `a l'Energie Atomique, 94250 Villeneuve Saint Georges cedex, France, email :despres@limeil.cea.fr Unit de recherche INRIA Rocquencourt Domaine de Voluceau, Rocquencourt, BP 105, 78153 LE CHESNAY Cedex (France) Tlphone : (33 1) 39 63 55 11 Tlcopie : (33 1) 39 63 53 30 Une m'ethode de d'ecomposition de domaine pour l"equation de Hem...
NONCONFORMING GALERKIN METHODS BASED ON QUADRILATERAL ELEMENTS FOR SECOND ORDER ELLIPTIC PROBLEMS
, 1999
"... Loworder nonconforming Galerkin methods will be analyzed for secondorder elliptic equations subjected to Robin, Dirichlet, or Neumann boundary conditions. Both simplicial and rectangular elements will be considered in two and three dimensions. The simplicial elements will be based on P1, as for ..."
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Cited by 28 (10 self)
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Loworder nonconforming Galerkin methods will be analyzed for secondorder elliptic equations subjected to Robin, Dirichlet, or Neumann boundary conditions. Both simplicial and rectangular elements will be considered in two and three dimensions. The simplicial elements will be based on P1, as for conforming elements; however, it is necessary to introduce new elements in the rectangular case. Optimal order error estimates are demonstrated in all cases with respect to a broken norm in H1(Ω) and in the Neumann and Robin cases in L²(Ω).
F.: On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs. Computational Geoscience 1:155–190
, 1997
"... Abstract. We present a new, naturally parallelizable, accurate numerical method for the solution of transportdominated diusion processes in heterogeneous porous media. For the discretization in time of one of the governing partial dierential equations, we introduce a new characteristicsbased proce ..."
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Cited by 22 (6 self)
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Abstract. We present a new, naturally parallelizable, accurate numerical method for the solution of transportdominated diusion processes in heterogeneous porous media. For the discretization in time of one of the governing partial dierential equations, we introduce a new characteristicsbased procedure which is mass conservative, the modied method of characteristics with adjusted advection (MMOCAA). Hybridized mixed nite elements are used for the spatial discretization of the equations and a new stripbased domain decomposition procedure is applied towards the solution of the resulting algebraic problems. We consider as a model problem the twophase immiscible displacement in petroleum reservoirs. A very detailed description of the numerical method is presented. Following that, numerical experiments are presented illustrating the important features of the new method and comparing computed results with ones derived from previous, related techniques. 1
Schwarz methods over the course of time
 Electronic Transactions on Numerical Analysis
, 2008
"... To the memory of Gene Golub, our leader and friend. Abstract. Schwarz domain decomposition methods are the oldest domain decomposition methods. They were invented by Hermann Amandus Schwarz in 1869 as an analytical tool to rigorously prove results obtained by Riemann through a minimization principle ..."
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Cited by 15 (2 self)
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To the memory of Gene Golub, our leader and friend. Abstract. Schwarz domain decomposition methods are the oldest domain decomposition methods. They were invented by Hermann Amandus Schwarz in 1869 as an analytical tool to rigorously prove results obtained by Riemann through a minimization principle. Renewed interest in these methods was sparked by the arrival of parallel computers, and variants of the method have been introduced and analyzed, both at the continuous and discrete level. It can be daunting to understand the similarities and subtle differences between all the variants, even for the specialist. This paper presents Schwarz methods as they were developed historically. From quotes by major contributors over time, we learn about the reasons for similarities and subtle differences between continuous and discrete variants. We also formally prove at the algebraic level equivalence and/or nonequivalence among the major variants for very general decompositions and many subdomains. We finally trace the motivations that led to the newest class called optimized Schwarz methods, illustrate how they can greatly enhance the performance of the solver, and show why one has to be cautious when testing them numerically. Key words. Alternating and parallel Schwarz methods, additive, multiplicative and restricted additive Schwarz methods, optimized Schwarz methods. AMS subject classifications. 65F10, 65N22.
A Domain Decomposition Method for Helmholtz Scattering Problems
 Ninth International Conference on Domain Decomposition Methods
, 1998
"... Introduction We present a study of iterative nonoverlapping domain decomposition methods (DDMs) for the harmonic scattering wave equation in the 3D case. We introduce some new nonlocal transmission conditions at subdomain interfaces in order to obtain an exponential rate of convergence. This work i ..."
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Cited by 12 (1 self)
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Introduction We present a study of iterative nonoverlapping domain decomposition methods (DDMs) for the harmonic scattering wave equation in the 3D case. We introduce some new nonlocal transmission conditions at subdomain interfaces in order to obtain an exponential rate of convergence. This work is a natural continuation of the work by Despres [Des91]. We present numerical results for a mixed finite element approximation. The parallel performance of the method on a tightly coupled machine and a loosely connected network is also shown. 2 Domain decomposition methods A model problem We study the scattering scalar Helmholtz equation in three dimensions. Let\Omega ae IR 3 be a bounded domain, \Gamma its boundary, and n the outgoing normal to \Gamma. The problem to solve is: 8 ? ? ? ! ? ? ? : (a) \Gammar( 1 ru) \Gamma<F1
Coupling of a NonOverlapping Domain Decomposition Method for a Nodal Finite . . .
, 2000
"... Nonoverlapping domain decomposition techniques are used both to solve the finite element equations and to couple them with a boundary element method. A suitable approach dealing with finite element nodes common to more than two subdomains, the socalled crosspoints, endows the method with the foll ..."
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Cited by 10 (3 self)
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Nonoverlapping domain decomposition techniques are used both to solve the finite element equations and to couple them with a boundary element method. A suitable approach dealing with finite element nodes common to more than two subdomains, the socalled crosspoints, endows the method with the following advantages. It yields a robust and efficient procedure to solve the equations resulting from the discretization process. Only small size finite element linear systems and a dense linear system related to a simple boundary integral equation are solved at each iteration and each of them can be solved in a stable way. We also show how to choose the parameter definining the augmented local matrices in order to improve the convergence. Several numerical simulations in 2D and 3D validating the treatment of the crosspoints and illustrating the strategy to accelerate the iterative procedure are presented.
Additive Schwarz Methods with Nonreflecting Boundary Conditions for the Parallel Computation of Helmholtz Problems
, 1998
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A Nonconforming Mixed Finite Element Method For Maxwell's Equations
"... this paper is to present a numerical procedure to determine the scattered electromagnetic elds induced inside the earth when a plane electromagnetic wave arrives normally to the earth's surface; the earth is modelled as a horizontallylayered medium containing arbitrarily shaped conductivity an ..."
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Cited by 4 (1 self)
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this paper is to present a numerical procedure to determine the scattered electromagnetic elds induced inside the earth when a plane electromagnetic wave arrives normally to the earth's surface; the earth is modelled as a horizontallylayered medium containing arbitrarily shaped conductivity anomalies. The numerical scheme is a new nonconforming mixed nite element procedure that can be hybridized in a manner leading to a domaindecomposition iterative technique to solve the algebraic equations associated with the procedure. Numerical methods to solve the direct problem in magnetotellurics have been proposed previously by several authors. In a classical work by P. E. Wannamaker et al. 27 , a nite element method was employed to solve the twodimensional scattering problem formulated as the timeharmonic Maxwell's equations considered as a set of two secondorder elliptic equations. Their method requires the calculation of derivatives of the conductivity coecient, introducing unnecessary numerical complexity. A moving nite element procedure to solve the twodimensional magnetotelluric problem was presented by B. Travis et al. 26 . A nite dierence procedure for threedimensional magnetotellurics was presented by R. L. Mackie et al. 14 . Finite dierence algorithms to solve Maxwell's equations in the timedomain have been widely used in electrical engineering applications, with the bestknown procedure being due to K. Yee 28 . A convergence analysis for Yee's scheme was given by P. Monk 19 , who also discussed nite element procedures for Maxwell's equations in two and three dimensions in several papers (see, e.g., 15;16;17;18 ). A collection of mixed nite element methods for twodimensional magnetotellurics has been recently presented in 21;22 . In this...
An analysis of the BEMFEM nonoverlapping domain decomposition method for a scattering problem
, 2005
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Fine Tuning Interface Relaxation Methods for Elliptic Differential Equations
, 1998
"... Two simple interface relaxation techniques for solving elliptic differential equations are considered. Their theoretical analysis is carried out at the differential level is carried out and "optimal" relaxation parameters are obtained for 1dimensional model problems. A comprehensive exper ..."
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Cited by 3 (1 self)
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Two simple interface relaxation techniques for solving elliptic differential equations are considered. Their theoretical analysis is carried out at the differential level is carried out and "optimal" relaxation parameters are obtained for 1dimensional model problems. A comprehensive experimental numerical study is also presented.