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A new cement to glue nonconforming grids with Robin interface conditions: the finite element case
, 2007
"... We design and analyze a new nonconforming domain decomposition method based on Schwarz type approaches that allows for the use of Robin interface conditions on nonconforming grids. The method is proven to be well posed, and the iterative solver to converge. The error analysis is performed in 2D p ..."
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Cited by 14 (7 self)
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We design and analyze a new nonconforming domain decomposition method based on Schwarz type approaches that allows for the use of Robin interface conditions on nonconforming grids. The method is proven to be well posed, and the iterative solver to converge. The error analysis is performed in 2D piecewise polynomials of low and high order and extended in 3D for P1 elements. Numerical results in 2D illustrate the new method.
SecondOrder Transmission Conditions for the Helmholtz Equation
 Ninth International Conference on Domain Decomposition Methods
, 1998
"... this paper a set of secondorder Robintype transmission conditions is proposed. The new transmission conditions significantly improve the rate of convergence of the iterates. After a brief overview of the general problem, including a general domain decomposition formulation, the specific model with ..."
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Cited by 3 (0 self)
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this paper a set of secondorder Robintype transmission conditions is proposed. The new transmission conditions significantly improve the rate of convergence of the iterates. After a brief overview of the general problem, including a general domain decomposition formulation, the specific model with secondorder Robintype transmission conditions is presented. A numerical example is used to demonstrate the benefits of this method. The discussion in this paper is intended to motivate the new algorithm and illustrate the type of improvements that are possible. A full description of the method with the accompanying theory is under development. 2 General Problem Many applications in electromagnetics, acoustics, and elasticity require the solution of a wave equation on an unbounded domain. A number of methods have been created for the reduction of the problem to a bounded domain. A common approach is to truncate the exterior domain and impose an appropriate boundary condition on the artificial boundary. The exact "radiation" boundary condition (RBC) is nonlocal (in both space Ninth International Conference on Domain Decomposition Methods Editor Petter E. Bjrstad, Magne S. Espedal and David E. Keyes c fl1998 DDM.org TRANSMISSION CONDITIONS FOR THE HELMHOLTZ EQUATION 435 and time); numerous spatially local approximate RBCs have also been developed (see, e.g., [Giv91]). Interest in domain decomposition methods for the solution of these problems arises from the fact that the direct solution of realistic scattering problems require the solution of large, sparse, complexvalued systems of linear equations. Domain decomposition methods are employed to create an iterative method requiring the direct solution of related problems on a small subdomain, typically a single biquad...
A NEW INTERFACE CEMENT EQUILIBRATED MORTAR (NICEM) METHOD WITH ROBIN INTERFACE CONDITIONS: THE P1 FINITE ELEMENT CASE
 MATHEMATICAL MODELS AND METHODS IN APPLIED SCIENCES
, 2012
"... We design and analyze a new nonconforming domain decomposition method, named the NICEM method, based on Schwarz type approaches that allows for the use of Robin interface conditions on nonconforming grids. The method is proven to be well posed. The error analysis is performed in 2D and in 3D for ..."
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Cited by 3 (1 self)
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We design and analyze a new nonconforming domain decomposition method, named the NICEM method, based on Schwarz type approaches that allows for the use of Robin interface conditions on nonconforming grids. The method is proven to be well posed. The error analysis is performed in 2D and in 3D for P1 elements. Numerical results in 2D illustrate the new method.
MCGS: A Modified Conjugate Gradient Squared Algorithm for Nonsymmetric Linear Systems
"... Abstract. The conjugate gradient squared CGS algorithm is a Krylov subspace algorithm that can be Ž . used to obtain fast solutions for linear systems Ax s b with complex nonsymmetric, very large, and Ž . very sparse coefficient matrices A . By considering electromagnetic scattering problems as exa ..."
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Abstract. The conjugate gradient squared CGS algorithm is a Krylov subspace algorithm that can be Ž . used to obtain fast solutions for linear systems Ax s b with complex nonsymmetric, very large, and Ž . very sparse coefficient matrices A . By considering electromagnetic scattering problems as examples, a study of the performance and scalability of this algorithm on two MIMD machines is presented. A Ž . modified CGS MCGS algorithm, where the synchronization overhead is effectively reduced by a factor of two, is proposed in this paper. This is achieved by changing the computation sequence in the CGS algorithm. Both experimental and theoretical analyses are performed to investigate the impact of this modification on the overall execution time. From the theoretical and experimental analysis it is found that CGS is faster than MCGS for smaller number of processors and MCGS outperforms CGS as the number of processors increases. Based on this observation, a set of algorithms approach is proposed, Ž . where either CGS or MGS is selected depending on the values of the dimension of the A matrix N Ž . and number of processors P . The set approach provides an algorithm that is more scalable than either the CGS or MCGS algorithms. The experiments performed on a 128processor mesh Intel Paragon and on a 16processor IBM SP2 with multistage network indicate that MCGS is approximately 20% faster than CGS.
MCGS: A Modified Conjugate Gradient Squared Algorithm for Nonsymmetric Linear Systems
"... . The conjugate gradient squared (CGS) algorithm is a Krylov subspace algorithm that can be used to obtain fast solutions for linear systems (Ax = b) with complex nonsymmetric, very large, and very sparse coefficient matrices (A). By considering electromagnetic scattering problems as examples, a stu ..."
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. The conjugate gradient squared (CGS) algorithm is a Krylov subspace algorithm that can be used to obtain fast solutions for linear systems (Ax = b) with complex nonsymmetric, very large, and very sparse coefficient matrices (A). By considering electromagnetic scattering problems as examples, a study of the performance and scalability of this algorithm on two MIMD machines is presented. A modified CGS (MCGS) algorithm, where the synchronization overhead is effectively reduced by a factor of two, is proposed in this paper. This is achieved by changing the computation sequence in the CGS algorithm. Both experimental and theoretical analyses are performed to investigate the impact of this modification on the overall execution time. From the theoretical and experimental analysis it is found that CGS is faster than MCGS for smaller number of processors and MCGS outperforms CGS as the number of processors increases. Based on this observation, a set of algorithms approach is proposed, where ...
MCGS: A Modi ed Conjugate Gradient Squared Algorithm for Nonsymmetric Linear Systems
"... Abstract. The conjugate gradient squared (CGS) algorithm is a Krylov subspace algorithm that can be used to obtain fast solutions for linear systems (Ax = b) with complex nonsymmetric, very large, and very sparse coe cient matrices (A). By considering electromagnetic scattering problems as examples, ..."
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Abstract. The conjugate gradient squared (CGS) algorithm is a Krylov subspace algorithm that can be used to obtain fast solutions for linear systems (Ax = b) with complex nonsymmetric, very large, and very sparse coe cient matrices (A). By considering electromagnetic scattering problems as examples, a study of the performance and scalability of this algorithm on two MIMD machines is presented. A modi ed CGS (MCGS) algorithm, where the synchronization overhead is e ectively reduced by a factor of two, is proposed in this paper. This is achieved by changing the computation sequence in the CGS algorithm. Both experimental and theoretical analyses are performed to investigate the impact of this modi cation on the overall execution time. From the theoretical and experimental analysis it is found that CGS is faster than MCGS for smaller number of processors and MCGS outperforms CGS as the number of processors increases. Based on this observation, a set of algorithms approach is proposed, where either CGS or MCGS is selected depending on the values of the dimension of the A matrix (N) and number of processors (P). The set approach provides an algorithm that is more scalable than either the CGS or MCGS algorithms. The experiments performed on a 128processor mesh Intel Paragon and on a 16processor IBM SP2 with multistage network indicate that MCGS is approximately 20 % faster than CGS.
Symmetrized Method with Optimized SecondOrder Conditions for the Helmholtz Equation
"... A schwarz type domain decomposition method for the Helmholtz equation is considered. The interface conditions involve second order tangential derivatives which are optimized (OO2, Optimized Order 2) for a fast convergence. The substructured form of the algorithm is symmetrized so that the symmetric ..."
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A schwarz type domain decomposition method for the Helmholtz equation is considered. The interface conditions involve second order tangential derivatives which are optimized (OO2, Optimized Order 2) for a fast convergence. The substructured form of the algorithm is symmetrized so that the symmetricQMR al
ROBIN SCHWARZ ALGORITHM FOR THE NICEM METHOD: THE PQ FINITE ELEMENT CASE
"... Abstract. In [16, 25] we proposed a new nonconforming domain decomposition paradigm, the New Interface Cement Equilibrated Mortar (NICEM) method, based on Schwarz type methods that allows for the use of Robin interface conditions on nonconforming grids. The error analysis was done for P1 finite el ..."
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Abstract. In [16, 25] we proposed a new nonconforming domain decomposition paradigm, the New Interface Cement Equilibrated Mortar (NICEM) method, based on Schwarz type methods that allows for the use of Robin interface conditions on nonconforming grids. The error analysis was done for P1 finite elements, in 2D and 3D. In this paper, we provide new numerical analysis results that allow to extend this error analysis in 2D for piecewise polynomials of higher order and also prove the convergence of the iterative algorithm in all these cases.