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Journal articles on the topic 'Linear equations; Schwarz methods'

Author: Grafiati

Published: 4 June 2021

Last updated: 8 February 2022

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1

Antonietti, Paola F., Blanca Ayuso de Dios, Susanne C. Brenner, and Li-yeng Sung. "Schwarz Methods for a Preconditioned WOPSIP Method for Elliptic Problems." Computational Methods in Applied Mathematics 12, no. 3 (2012): 241–72. http://dx.doi.org/10.2478/cmam-2012-0021.

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Abstract We propose and analyze several two-level non-overlapping Schwarz methods for a preconditioned weakly over-penalized symmetric interior penalty (WOPSIP) discretization of a second order boundary value problem. We show that the preconditioners are scalable and that the condition number of the resulting preconditioned linear systems of equations is independent of the penalty parameter and is of order H/h, where H and h represent the mesh sizes of the coarse and fine partitions, respectively. Numerical experiments that illustrate the performance of the proposed two-level Schwarz methods are also presented.

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2

Nagid, Nabila, and Hassan Belhadj. "New approach for accelerating the nonlinear Schwarz iterations." Boletim da Sociedade Paranaense de Matemática 38, no. 4 (2019): 51–69. http://dx.doi.org/10.5269/bspm.v38i4.37018.

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The vector Epsilon algorithm is an effective extrapolation method used for accelerating the convergence of vector sequences. In this paper, this method is used to accelerate the convergence of Schwarz iterative methods for stationary linear and nonlinear partial differential equations (PDEs). The vector Epsilon algorithm is applied to the vector sequences produced by additive Schwarz (AS) or restricted additive Schwarz (RAS) methods after discretization. Some convergence analysis is presented, and several test-cases of analytical problems are performed in order to illustrate the interest of such algorithm. The obtained results show that the proposed algorithm yields much faster convergence than the classical Schwarz iterations.

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3

Antoine, Xavier, Fengji Hou, and Emmanuel Lorin. "Asymptotic estimates of the convergence of classical Schwarz waveform relaxation domain decomposition methods for two-dimensional stationary quantum waves." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 4 (2018): 1569–96. http://dx.doi.org/10.1051/m2an/2017048.

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This paper is devoted to the analysis of convergence of Schwarz Waveform Relaxation (SWR) domain decomposition methods (DDM) for solving the stationary linear and nonlinear Schrödinger equations by the imaginary-time method. Although SWR are extensively used for numerically solving high-dimensional quantum and classical wave equations, the analysis of convergence and of the rate of convergence is still largely open for linear equations with variable coefficients and nonlinear equations. The aim of this paper is to tackle this problem for both the linear and nonlinear Schrödinger equations in the two-dimensional setting. By extending ideas and concepts presented earlier [X. Antoine and E. Lorin, Numer. Math. 137 (2017) 923–958] and by using pseudodifferential calculus, we prove the convergence and determine some approximate rates of convergence of the two-dimensional Classical SWR method for two subdomains with smooth boundary. Some numerical experiments are also proposed to validate the analysis.

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4

Wu, Shu-Lin. "Schwarz Waveform Relaxation for Heat Equations with Nonlinear Dynamical Boundary Conditions." Abstract and Applied Analysis 2013 (2013): 1–7. http://dx.doi.org/10.1155/2013/474608.

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We are interested in solving heat equations with nonlinear dynamical boundary conditions by using domain decomposition methods. In the classical framework, one first discretizes the time direction and then solves a sequence of state steady problems by the domain decomposition method. In this paper, we consider the heat equations at spacetime continuous level and study a Schwarz waveform relaxation algorithm for parallel computation purpose. We prove the linear convergence of the algorithm on long time intervals and show how the convergence rate depends on the size of overlap and the nonlinearity of the nonlinear boundary functions. Numerical experiments are presented to verify our theoretical conclusions.

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5

Antoine, X., and E. Lorin. "An analysis of Schwarz waveform relaxation domain decomposition methods for the imaginary-time linear Schrödinger and Gross-Pitaevskii equations." Numerische Mathematik 137, no. 4 (2017): 923–58. http://dx.doi.org/10.1007/s00211-017-0897-3.

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6

Lapenta, Giovanni, and Wei Jiang. "Implicit Temporal Discretization and Exact Energy Conservation for Particle Methods Applied to the Poisson–Boltzmann Equation." Plasma 1, no. 2 (2018): 242–58. http://dx.doi.org/10.3390/plasma1020021.

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We report on a new multiscale method approach for the study of systems with wide separation of short-range forces acting on short time scales and long-range forces acting on much slower scales. We consider the case of the Poisson–Boltzmann equation that describes the long-range forces using the Boltzmann formula (i.e., we assume the medium to be in quasi local thermal equilibrium). We develop a new approach where fields and particle information (mediated by the equations for their moments) are solved self-consistently. The new approach is implicit and numerically stable, providing exact energy conservation. We test different implementations that all lead to exact energy conservation. The new method requires the solution of a large set of non-linear equations. We consider three solution strategies: Jacobian Free Newton Krylov, an alternative, called field hiding which is based on hiding part of the residual calculation and replacing them with direct solutions and a Direct Newton Schwarz solver that considers a simplified, single, particle-based Jacobian. The field hiding strategy proves to be the most efficient approach.

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7

Гурьева, Я. Л., and В. П. Ильин. "On acceleration technologies of parallel decomposition methods." Numerical Methods and Programming (Vychislitel'nye Metody i Programmirovanie), no. 1 (April 2, 2015): 146–54. http://dx.doi.org/10.26089/nummet.v16r115.

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Одним из главных препятствий масштабированному распараллеливанию алгебраических методов декомпозиции для решения сверхбольших разреженных систем линейных алгебраических уравнений (СЛАУ) является замедление скорости сходимости аддитивного итерационного алгоритма Шварца в подпространствах Крылова при увеличении количества подобластей. Целью настоящей статьи является сравнительный экспериментальный анализ различных приeмов ускорения итераций: параметризованное пересечение подобластей, использование специальных интерфейсных условий на границах смежных подобластей, а также применение грубосеточной коррекции (агрегации, или редукции) исходной СЛАУ для построения дополнительного предобусловливателя. Распараллеливание алгоритмов осуществляется на двух уровнях программными средствами для распределeнной и общей памяти. Тестовые СЛАУ получаются при помощи конечно-разностных аппроксимаций задачи Дирихле для диффузионно-конвективного уравнения с различными значениями конвективных коэффициентов на последовательности сгущающихся сеток. One of the main obstacles to the scalable parallelization of the algebraic decomposition methods for solving large sparse systems of linear algebraic equations consists in slowing the convergence rate of the additive iterative Schwarz algorithm in the Krylov subspaces when the number of subdomains increases. The aim of this paper is a comparative experimental analysis of various ways to accelerate the iterations: a parametrized intersection of subdomains, the usage of interface conditions at the boundaries of adjacent subdomains, and the application of a coarse grid correction (aggregation, or reduction) for the original linear system to build an additional preconditioner. The parallelization of algorithms is performed on two levels by programming tools for the distributed and shared memory. The benchmark linear systems under study are formed using the finite difference approximations of the Dirichlet problem for the diffusion-convection equation with various values of the convection coefficients and on a sequence of condensing grids.

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8

Zhou, H., and H. A. A. Tchelepi. "Two-Stage Algebraic Multiscale Linear Solver for Highly Heterogeneous Reservoir Models." SPE Journal 17, no. 02 (2012): 523–39. http://dx.doi.org/10.2118/141473-pa.

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Summary An efficient Two-Stage Algebraic Multiscale Solver (TAMS) that converges to the fine-scale solution is described. The first (global) stage is a multiscale solution obtained algebraically for the given fine-scale problem. In the second stage, a local preconditioner, such as the Block ILU (BILU), or the Additive Schwarz (AS) method is used. Spectral analysis shows that the multiscale solution step captures the low-frequency parts of the error spectrum quite well, while the local preconditioner represents the high-frequency components accurately. Combining the two stages in an iterative scheme results in efficient treatment of all the error components associated with the fine-scale problem. TAMS is shown to converge to the reference fine-scale solution. Moreover, the eigenvalues of the TAMS iteration matrix show significant clustering, which is favorable for Krylov-based methods. Accurate solution of the nonlinear saturation equations (i.e., transport problem) requires having locally conservative velocity fields. TAMS guarantees local mass conservation by concluding the iterations with a multiscale finite-volume step. We demonstrate the performance of TAMS using several test cases with strong permeability heterogeneity and large-grid aspect ratios. Different choices in the TAMS algorithm are investigated, including the Galerkin and finite-volume restriction operators, as well as the BILU and AS preconditioners for the second stage. TAMS for the elliptic flow problem is comparable to state-of-the-art algebraic multigrid methods, which are in wide use. Moreover, the computational time of TAMS grows nearly linearly with problem size.

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9

Kong, Fande, and Xiao-Chuan Cai. "Scalability study of an implicit solver for coupled fluid-structure interaction problems on unstructured meshes in 3D." International Journal of High Performance Computing Applications 32, no. 2 (2016): 207–19. http://dx.doi.org/10.1177/1094342016646437.

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Fluid-structure interaction (FSI) problems are computationally very challenging. In this paper we consider the monolithic approach for solving the fully coupled FSI problem. Most existing techniques, such as multigrid methods, do not work well for the coupled system since the system consists of elliptic, parabolic and hyperbolic components all together. Other approaches based on direct solvers do not scale to large numbers of processors. In this paper, we introduce a multilevel unstructured mesh Schwarz preconditioned Newton–Krylov method for the implicitly discretized, fully coupled system of partial differential equations consisting of incompressible Navier–Stokes equations for the fluid flows and the linear elasticity equation for the structure. Several meshes are required to make the solution algorithm scalable. This includes a fine mesh to guarantee the solution accuracy, and a few isogeometric coarse meshes to speed up the convergence. Special attention is paid when constructing and partitioning the preconditioning meshes so that the communication cost is minimized when the number of processor cores is large. We show numerically that the proposed algorithm is highly scalable in terms of the number of iterations and the total compute time on a supercomputer with more than 10,000 processor cores for monolithically coupled three-dimensional FSI problems with hundreds of millions of unknowns.

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10

Phillips, Peter C. B., and Werner Ploberger. "Posterior Odds Testing for a Unit Root with Data-Based Model Selection." Econometric Theory 10, no. 3-4 (1994): 774–808. http://dx.doi.org/10.1017/s026646660000877x.

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The Kalman filter is used to derive updating equations for the Bayesian data density in discrete time linear regression models with stochastic regressors. The implied “Bayes model” has time varying parameters and conditionally heterogeneous error variances. A σ-finite Bayes model measure is given and used to produce a new-model-selection criterion (PIC) and objective posterior odds tests for sharp null hypotheses like the presence of a unit root. This extends earlier work by Phillips and Ploberger [18]. Autoregressive-moving average (ARMA) models are considered, and a general test of trend-stationarity versus difference stationarity is developed in ARMA models that allow for automatic order selection of the stochastic regressors and the degree of the deterministic trend. The tests are completely consistent in that both type I and type II errors tend to zero as the sample size tends to infinity. Simulation results and an empirical application are reported. The simulations show that the PIC works very well and is generally superior to the Schwarz BIC criterion, even in stationary systems. Empirical application of our methods to the Nelson-Plosser [11] series show that three series (unemployment, industrial production, and the money stock) are level- or trend-stationary. The other eleven series are found to be stochastically nonstationary.

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11

Colli Franzone, Piero, Luca F. Pavarino, and Simone Scacchi. "Bioelectrical effects of mechanical feedbacks in a strongly coupled cardiac electro-mechanical model." Mathematical Models and Methods in Applied Sciences 26, no. 01 (2015): 27–57. http://dx.doi.org/10.1142/s0218202516500020.

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The aim of this work is to investigate by means of numerical simulations the effects of myocardial deformation due to muscle contraction on the bioelectrical activity of the cardiac tissue. The three-dimensional electro-mechanical model considered consists of the following four components: the quasi-static orthotropic finite elasticity equations for the deformation of the cardiac tissue; the active tension model for the intracellular calcium dynamics and cross-bridge binding; the orthotropic Bidomain model for the electrical current flow through the tissue; the membrane model of the cardiac myocyte, including stretch-activated currents (I SAC ). In order to properly take into account cardiac mechanical feedbacks, the electrical current flow is described in a strongly coupled framework by the Bidomain model on the deformed tissue. We then derive a novel formulation of the Bidomain model in the reference configuration, with complete mechanical feedbacks affecting not only the conductivity tensors but also a convective term depending on the velocity of the deformation. The numerical simulations are based on our finite element parallel solver, which employs both Multilevel Additive Schwarz preconditioners for the solution of linear systems arising from the discretization of the Bidomain equations and Newton–Krylov-Algebraic Multigrid methods for the solution of nonlinear systems arising from the discretization of the finite elasticity equations. The results have shown that: (i) the I SAC current prolongs action potential duration (APD) of about 10–15 ms; (ii) the inclusion into the model of both I SAC current and the convective term reduces the dispersion of repolarization of about 7% (from 139 to 129 ms) and increases the dispersion of APD about three times (from 13 to 45 ms). These effects indicate that mechanical feedbacks might influence arrhythmogenic mechanisms when combined with pathological substrates.

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12

JAKOBSEN, ESPEN ROBSTAD. "ON THE RATE OF CONVERGENCE OF APPROXIMATION SCHEMES FOR BELLMAN EQUATIONS ASSOCIATED WITH OPTIMAL STOPPING TIME PROBLEMS." Mathematical Models and Methods in Applied Sciences 13, no. 05 (2003): 613–44. http://dx.doi.org/10.1142/s0218202503002660.

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We provide estimates on the rate of convergence for approximation schemes for Bellman equations associated with optimal stopping of controlled diffusion processes. These results extend (and slightly improve) the recent results by Barles & Jakobsen to the more difficult time-dependent case. The added difficulties are due to the presence of boundary conditions (initial conditions!) and the new structure of the equation which is now a parabolic variational inequality. The method presented is purely analytic and rather general and is based on earlier work by Krylov and Barles & Jakobsen. As applications we consider so-called control schemes based on the dynamic programming principle and finite difference methods (though not in the most general case). In the optimal stopping case these methods are similar to the Brennan & Schwartz scheme. A simple observation allows us to obtain the optimal rate 1/2 for the finite difference methods, and this is an improvement over previous results by Krylov and Barles & Jakobsen. Finally, we present an idea that allows us to improve all the above-mentioned results in the linear case. In particular, we are able to handle finite difference methods with variable diffusion coefficients without the reduction of order of convergence observed by Krylov in the nonlinear case.

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13

Stankevich, I. V., and P. S. Aronov. "Mathematical Modeling of the Contact Interaction of Two Elastic Bodies Using the Mortar Method." Mathematics and Mathematical Modeling, no. 3 (August 3, 2018): 26–44. http://dx.doi.org/10.24108/mathm.0318.0000112.

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The article discusses the development of an algorithm for solving contact problems of elasticity theory. Solving such problems is often associated with necessity of using mismatched grids. Their joining can be carried out both with the help of iterative procedures that form the so-called Schwarz alternating methods, and with the help of the Lagrange multipliers method or the penalty method. The algorithm constructed in the article uses the mortar method for matching the finite elements on the contact line. All these methods of joining the grids make it possible to ensure continuity of displacements and stresses near the contact line. However, one of the main advantages of the mortar method is the possibility of independent choice of different types of finite elements and form functions both on both boundaries of two bodies on the contact line, and when integrating along it. The application of this method in conjunction with the classical formulation of the finite element method based on the minimization of the Lagrange functional leads to a system of linear algebraic equations with a saddle point. The article discusses in detail its numerical solution based on the modified symmetric successive upper relaxation method.The results of the constructed algorithm are demonstrated on three test contact problems. They analyze the stress-strain state of differently loaded contacting two-dimensional plates. The examples considered show that continuity of the displacements of displacements and stresses is preserved near the contact line. The versatility of the developed algorithm leaves the possibility of further analysis of the effectiveness of the mortar method using different types of finite elements and form functions.

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14

Gnatyuk, M. A., and V. M. Morozov. "An integral equation technique for the analysis of phased array antenna with matching step discontinuities." Journal of Physics and Electronics 26, no. 2 (2018): 101–6. http://dx.doi.org/10.15421/331833.

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Applying the integral equation method of overlapping partial domains and the Schwartz alternating method to solving an electromagnetic wave diffraction problem is considered in this paper. The infinite rectangular waveguide phased array antenna scanning in H plane which waveguides have step matching discontinuities is represented. The whole field definition domain is sliced into three overlapping partial domains. The system of integral representations for unknown Ey components of the electrical field vector in each domain is set up using Greenʼs functions. Unknown functions in each domain are presented as orthogonal eigenfunction series. Using Galerkinʼs procedure, the system of integral representations is reduced to the system of linear equations for unknown expansion coefficients. For Schwartz method the system of integral representation is solved using iterative methods. The dependences of the reflection coefficient magnitude and phase on the value of scan angle are obtained. The comparison of obtained results for particular cases with known ones is performed.

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15

Dolean, V., M. J. Gander, and L. Gerardo-Giorda. "Optimized Schwarz Methods for Maxwell's Equations." SIAM Journal on Scientific Computing 31, no. 3 (2009): 2193–213. http://dx.doi.org/10.1137/080728536.

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16

Sanabria Malagón, Camilo. "Schwarz maps of algebraic linear ordinary differential equations." Journal of Differential Equations 263, no. 11 (2017): 7123–40. http://dx.doi.org/10.1016/j.jde.2017.08.002.

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17

Migliorati, Giovanni, and Alfio Quarteroni. "Multilevel Schwarz methods for elliptic partial differential equations." Computer Methods in Applied Mechanics and Engineering 200, no. 25-28 (2011): 2282–96. http://dx.doi.org/10.1016/j.cma.2011.03.017.

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18

Aksoy, Ü., and A. O. Çelebi. "Schwarz problem for higher order linear equations in a polydisc." Complex Variables and Elliptic Equations 62, no. 10 (2017): 1558–69. http://dx.doi.org/10.1080/17476933.2016.1254627.

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19

Boulbrachene, Messaoud. "Finite Element Convergence Analysis of a Schwarz Alternating Method for Nonlinear Elliptic PDEs." Sultan Qaboos University Journal for Science [SQUJS] 24, no. 2 (2020): 109. http://dx.doi.org/10.24200/squjs.vol24iss2pp109-121.

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In this paper, we prove uniform convergence of the standard finite element method for a Schwarz alternating procedure for nonlinear elliptic partial differential equations in the context of linear subdomain problems and nonmatching grids. The method stands on the combination of the convergence of linear Schwarz sequences with standard finite element L-error estimate for linear problems.

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20

Miyajima, Keiichi, Artur Korniłowicz, and Yasunari Shidama. "Contracting Mapping on Normed Linear Space." Formalized Mathematics 20, no. 4 (2012): 291–301. http://dx.doi.org/10.2478/v10037-012-0035-8.

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Summary In this article, we described the contracting mapping on normed linear space. Furthermore, we applied that mapping to ordinary differential equations on real normed space. Our method is based on the one presented by Schwarz [29].

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21

Li, Shishun, Xinping Shao, and Xiao-Chuan Cai. "Multilevel Space-Time Additive Schwarz Methods for Parabolic Equations." SIAM Journal on Scientific Computing 40, no. 5 (2018): A3012—A3037. http://dx.doi.org/10.1137/17m113808x.

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22

Blayo, Eric, David Cherel, and Antoine Rousseau. "Towards Optimized Schwarz Methods for the Navier–Stokes Equations." Journal of Scientific Computing 66, no. 1 (2015): 275–95. http://dx.doi.org/10.1007/s10915-015-0020-9.

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23

Toselli, Andrea. "Overlapping Schwarz methods for Maxwell's equations in three dimensions." Numerische Mathematik 86, no. 4 (2000): 733–52. http://dx.doi.org/10.1007/pl00005417.

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24

Guo, Guangbao. "Schwarz Methods for Quasi-Likelihood in Generalized Linear Models." Communications in Statistics - Simulation and Computation 37, no. 10 (2008): 2027–36. http://dx.doi.org/10.1080/03610910802311700.

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25

Dmitriev, М. О. "Determination of individual teleroentgenographic characteristics of the face profile in ukrainian young men and girls with orthognathic bite." Biomedical and Biosocial Anthropology, no. 32 (September 20, 2018): 28–34. http://dx.doi.org/10.31393/bba32-2018-04.

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Modern dentistry requires the definition of individualized values of teleroentgenographic indicators. To solve such problems, methods of regression and correlation analysis are increasingly used, which help to establish not only the existence of various relationships between the anatomical structures of the head and the parameters of the dento-jaw system, but also allow more accurately predict the change in the contour of soft facial tissue in response to orthodontic treatment. The purpose of the study is to develop mathematical models for the determination of individual teleroentgenographic characteristics of the facial soft tissues by studying the cephalometric indices of young men and women of Ukraine with normal occlusion and balanced faces and conducting a direct stepwise regression analysis. With the use of Veraviewepocs 3D device, Morita (Japan) from 38 young men (17 to 21 years of age) and 55 young women (aged from 16 to 20 years) with occlusal close to the orthognathic bite and balanced faces received side teleroentgenograms. The cephalometric analysis was performed using OnyxCeph³™ licensed software. Cephalometric points and measurements were made according to the recommendations of Downs W. B., Holdway R. A., McNamara J., Schwarz A. M., Schmuth G. P. F., Steiner C. C. and Tweed C. H. With the help of direct stepwise regression analysis, in the licensed package “Statistica 6.0”, regression models of individual teleroentgenographic characteristics of the profile of soft facial tissues were constructed. In young men with normal occlusion close to the orthognathic bite of 19 possible models, 11 were constructed with a determination coefficient from 0.638 to 0.930, and in young women – 12 models with a determination coefficient from 0.541 to 0.927. The conducted analysis of models showed that in young men most often the regression equations included – angle N_POG, parameters of which indicate a linear interjaw relation in the anterior-posterior direction (14.0%); angle GL_SNPOG, or index of convexity of the soft tissue profile (8.8%); MAX maxillary length (7.0%), and GL_SN_S index, which defines vertical correlations in the facial profile (5.3%). The young women most often models included – the angle N_POG (12.5%); angle GL_SNPOG (7.5%); soft tissue front angle P_OR_N (6.25%); the reference angle ML_NL and the profile angle T (by 5.0%); the angle AB_NPOG, the angle NBA_PTGN, which defines the direction of development of the mandible and the distance PN_A (3.75%). Thus, in the work with the help of the method of stepwise regression with inclusion, among Ukrainians of adolescence age, based on the characteristics of teleroentgenographic indicators, reliable models of individual teleroentgenographic characteristics of the profile of soft facial tissues were developed and analyzed.

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Martin, Véronique. "Schwarz Waveform Relaxation Algorithms for the Linear Viscous Equatorial Shallow Water Equations." SIAM Journal on Scientific Computing 31, no. 5 (2009): 3595–625. http://dx.doi.org/10.1137/070691450.

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27

Magoulès, Frédéric, Pascal Laurent-Gengoux, and Florent Pruvost. "Preconditioners for Schwarz relaxation methods applied to differential algebraic equations." International Journal of Computer Mathematics 91, no. 8 (2014): 1775–89. http://dx.doi.org/10.1080/00207160.2013.862524.

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28

Tran, Minh-Binh. "Overlapping optimized Schwarz methods for parabolic equations in $n$ dimensions." Proceedings of the American Mathematical Society 141, no. 5 (2012): 1627–40. http://dx.doi.org/10.1090/s0002-9939-2012-11522-9.

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29

Lui, S. H. "On Schwarz Alternating Methods for the Incompressible Navier--Stokes Equations." SIAM Journal on Scientific Computing 22, no. 6 (2001): 1974–86. http://dx.doi.org/10.1137/s1064827598347411.

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30

Klawonn, Axel, and Luca F. Pavarino. "Overlapping Schwarz methods for mixed linear elasticity and Stokes problems." Computer Methods in Applied Mechanics and Engineering 165, no. 1-4 (1998): 233–45. http://dx.doi.org/10.1016/s0045-7825(98)00059-0.

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31

Pauler, D. "The Schwarz criterion and related methods for normal linear models." Biometrika 85, no. 1 (1998): 13–27. http://dx.doi.org/10.1093/biomet/85.1.13.

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32

Sun, Zhe, Jinping Zeng, and Donghui Li. "Semismooth Newton Schwarz iterative methods for the linear complementarity problem." BIT Numerical Mathematics 50, no. 2 (2010): 425–49. http://dx.doi.org/10.1007/s10543-010-0261-9.

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33

Li, Shishun, Rongliang Chen, and Xinping Shao. "Parallel two-level space–time hybrid Schwarz method for solving linear parabolic equations." Applied Numerical Mathematics 139 (May 2019): 120–35. http://dx.doi.org/10.1016/j.apnum.2019.01.016.

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Beuchler, Sven, and Martin Purrucker. "Schwarz Type Solvers for -FEM Discretizations of Mixed Problems." Computational Methods in Applied Mathematics 12, no. 4 (2012): 369–90. http://dx.doi.org/10.2478/cmam-2012-0030.

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AbstractThis paper investigates the discretization of mixed variational formulation as, e.g., the Stokes problem by means of the hp-version of the finite element method. The system of linear algebraic equations is solved by the preconditioned Bramble-Pasciak conjugate gradient method. The development of an efficient preconditioner requires three ingredients, a preconditioner related to the components of the velocity modes, a preconditioner for the Schur complement related to the components of the pressure modes and a discrezation by a stable finite element pair which satisfies the discrete inf-sup-condition. The last condition is also important in order to obtain a stable discretization scheme. The preconditioner for the velocity modes is adapted from fast $hp$-FEM preconditioners for the potential equation. Moreover, we will prove that the preconditioner for the Schur complement can be chosen as a diagonal matrix if the pressure is discretized by discontinuous finite elements. We will prove that the system of linear algebraic equations can be solved in almost optimal complexity. This yields quasioptimal hp-FEM solvers for the Stokes problems and the linear elasticity problems. The latter are robust with respect to the contraction ratio ν. The efficiency of the presented solver is shown in several numerical examples.

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Wu, Shu-Lin, and Cheng-Ming Huang. "Quasi-optimized Schwarz methods for reaction diffusion equations with time delay." Journal of Mathematical Analysis and Applications 385, no. 1 (2012): 354–70. http://dx.doi.org/10.1016/j.jmaa.2011.06.052.

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Bouajaji, M. El, V. Dolean, M. J. Gander, and S. Lanteri. "Optimized Schwarz Methods for the Time-Harmonic Maxwell Equations with Damping." SIAM Journal on Scientific Computing 34, no. 4 (2012): A2048—A2071. http://dx.doi.org/10.1137/110842995.

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Jiang, Yao-Lin, and Hui Zhang. "Schwarz waveform relaxation methods for parabolic equations in space-frequency domain." Computers & Mathematics with Applications 55, no. 12 (2008): 2924–39. http://dx.doi.org/10.1016/j.camwa.2007.11.025.

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38

Lui, S. H. "On linear monotone iteration and Schwarz methods for nonlinear elliptic PDEs." Numerische Mathematik 93, no. 1 (2002): 109–29. http://dx.doi.org/10.1007/bf02679439.

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Schwandt, Hartmut. "Parallel interval Newton-like Schwarz methods for almost linear parabolic problems." Journal of Computational and Applied Mathematics 199, no. 2 (2007): 437–44. http://dx.doi.org/10.1016/j.cam.2005.07.042.

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Simoncini, V. "Computational Methods for Linear Matrix Equations." SIAM Review 58, no. 3 (2016): 377–441. http://dx.doi.org/10.1137/130912839.

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Dolean, Victorita, Martin J. Gander, and Erwin Veneros. "Asymptotic analysis of optimized Schwarz methods for maxwell’s equations with discontinuous coefficients." ESAIM: Mathematical Modelling and Numerical Analysis 52, no. 6 (2018): 2457–77. http://dx.doi.org/10.1051/m2an/2018041.

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Abstract:

Discretized time harmonic Maxwell’s equations are hard to solve by iterative methods, and the best currently available methods are based on domain decomposition and optimized transmission conditions. Optimized Schwarz methods were the first ones to use such transmission conditions, and this approach turned out to be so fundamentally important that it has been rediscovered over the last years under the name sweeping preconditioners, source transfer, single layer potential method and the method of polarized traces. We show here how one can optimize transmission conditions to take benefit from the jumps in the coefficients of the problem, when they are aligned with the subdomain interface, and obtain methods which converge for two subdomains in certain situations independently of the mesh size, which would not be possible without jumps in the coefficients.

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Stephan, Ernst P., Matthias Maischak, and Florian Leydecker. "Some schwarz methods for integral equations on surfaces-h and p versions." Computing and Visualization in Science 8, no. 3-4 (2005): 211–16. http://dx.doi.org/10.1007/s00791-005-0011-8.

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Liu, Cuiyu, and Chen-liang Li. "A Preconditioned Multisplitting and Schwarz Method for Linear Complementarity Problem." Journal of Applied Mathematics 2014 (2014): 1–6. http://dx.doi.org/10.1155/2014/519017.

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The preconditioner presented by Hadjidimos et al. (2003) can improve on the convergence rate of the classical iterative methods to solve linear systems. In this paper, we extend this preconditioner to solve linear complementarity problems whose coefficient matrix isM-matrix orH-matrix and present a multisplitting and Schwarz method. The convergence theorems are given. The numerical experiments show that the methods are efficient.

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Gander, Martin, Laurence Halpern, Frédéric Magoulès, and François-Xavier Roux. "Analysis of Patch Substructuring Methods." International Journal of Applied Mathematics and Computer Science 17, no. 3 (2007): 395–402. http://dx.doi.org/10.2478/v10006-007-0032-1.

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Analysis of Patch Substructuring MethodsPatch substructuring methods are non-overlapping domain decomposition methods like classical substructuring methods, but they use information from geometric patches reaching into neighboring subdomains, condensated on the interfaces, to enhance the performance of the method, while keeping it non-overlapping. These methods are very convenient to use in practice, but their convergence properties have not been studied yet. We analyze geometric patch substructuring methods for the special case of one patch per interface. We show that this method is equivalent to an overlapping Schwarz method using Neumann transmission conditions. This equivalence is obtained by first studying a new, algebraic patch method, which is equivalent to the classical Schwarz method with Dirichlet transmission conditions and an overlap corresponding to the size of the patches. Our results motivate a new method, the Robin patch method, which is a linear combination of the algebraic and the geometric one, and can be interpreted as an optimized Schwarz method with Robin transmission conditions. This new method has a significantly faster convergence rate than both the algebraic and the geometric one. We complement our results by numerical experiments.

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Pavarino, Luca F., and Elena Zampieri. "Overlapping Schwarz and Spectral Element Methods for Linear Elasticity and Elastic Waves." Journal of Scientific Computing 27, no. 1-3 (2006): 51–73. http://dx.doi.org/10.1007/s10915-005-9047-7.

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Blayo, Eric, Antoine Rousseau, and Manel Tayachi. "Boundary conditions and Schwarz waveform relaxation method for linear viscous Shallow Water equations in hydrodynamics." SMAI journal of computational mathematics 3 (September 14, 2017): 117–37. http://dx.doi.org/10.5802/smai-jcm.22.

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Frommer, Andreas, Reinhard Nabben, and Daniel B. Szyld. "Convergence of Stationary Iterative Methods for Hermitian Semidefinite Linear Systems and Applications to Schwarz Methods." SIAM Journal on Matrix Analysis and Applications 30, no. 2 (2008): 925–38. http://dx.doi.org/10.1137/080714038.

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Heuer, Norbert. "Additive schwarz methods for indefinite hypersingular integral equations in R3- the p-version." Applicable Analysis 72, no. 3-4 (1999): 411–37. http://dx.doi.org/10.1080/00036819908840750.

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de Gee, Maarten. "Linear Multistep Methods for Functional Differential Equations." Mathematics of Computation 48, no. 178 (1987): 633. http://dx.doi.org/10.2307/2007833.

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Chen, Yong-Lin. "Iterative methods for solving restricted linear equations." Applied Mathematics and Computation 86, no. 2-3 (1997): 171–84. http://dx.doi.org/10.1016/s0096-3003(96)00180-4.

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