1,720,992 research outputs found
Happy 25th Anniversary DDM! ... But How Fast Can the Schwarz Method Solve Your Logo?
No full text“Vous n’avez vraiment rien à faire”!1 This was the smiling reaction of Laurence Halpern when the first author told her about our wish to accurately estimate the convergence rate of the Schwarz method for the solution of the ddm logo2, see Figure 1 (left)
Unmapped Tent Pitching Schemes by Waveform Relaxation
No full textThe mapped tent pitching algorithm (MTP) is a very advanced domain decomposition strategy for the parallel solution of hyperbolic problems
Partition of unity methods for heterogeneous domain decomposition
No full textIn many applications, mathematical and numerical models involve simultaneously more than one single phenomenon. In this situation different equations are used in possibly overlapping subregions of the domain in order to approximate the physical model and obtain an efficient reduction of the computational cost. The coupling between the different equations must be carefully handled to guarantee accurate results. However in many cases, since the geometry of the overlapping subdomains is neither given a-priori nor characterized by coupling equations, a matching relation between the different equations is not available; see, e.g. Degond and Jin (SIAM J Numer Anal 42(6):2671–2687, 2005), Gander et al. (Numer Algorithm 73(1):167–195, 2016) and references therein. To overcome this problem, we introduce a new methodology that interprets the (unknown) decomposition of the domain by associating each subdomain to a partition of unity (membership) function. Then, by exploiting the feature of the partition of unity method developed in Babuska and Melenk (Int J Numer Methods Eng 40:727–758, 1996) and Griebel and Schweitzer (SIAM J Sci Comput 22(3):853–890, 2000), we define a new domain-decomposition strategy that can be easily embedded in infinite-dimensional optimization settings. This allows us to develop a new optimal control methodology that is capable to design coupling mechanisms between the different approximate equations. Numerical experiments demonstrate the efficiency of the proposed framework
Heterogeneous Optimized Schwarz Methods for Second Order Elliptic PDEs
Full text linkDue to their property of convergence in the absence of overlap, optimized Schwarz methods are the natural domain decomposition framework for heterogeneous problems, where the spatial decomposition is provided by the multiphysics of the phenomena. We study here heterogeneous problems which arise from the coupling of second order elliptic PDEs. Theoretical results and asymptotic formulas are proposed solving the corresponding min-max problems both for single and double sided optimizations, while numerical results confirm the effectiveness of our approach even when analytical conclusions are not available. Our analysis shows that optimized Schwarz methods do not suffer the heterogeneity, it is the opposite, they are faster the stronger the heterogeneity is. It is even possible to have h independent convergence choosing two independent Robin parameters. This property was proved for a Laplace equation with discontinuous coefficients, but only conjectured for more general couplings in [M. J. Gander and O. Dubois, Numer. Algorithms, 69 (2015), pp. 109-144]. Our study is completed by an application to a contaminant transport problem
Multilevel Optimized Schwarz Methods
No full textWe define a new two-level optimized Schwarz method (OSM), and we provide a convergence analysis both for overlapping and nonoverlapping decompositions. The two-level analysis suggests how to choose the optimized parameters. We also discuss an optimization procedure which relies only on the already studied one-level min-max problems, and we show that these two approaches are asymptotically equivalent. The two-level OSM has mesh independent convergence and it is scalable. We then generalize the two-level method defining a multilevel domain decomposition method which uses the OSM as a smoother. The main advantage of the method consists of its robustness and generality with respect to the equations under study. Thanks to the smoothing properties of the OSM, both with and without overlap, we can define a unique algorithm which can be applied to several equations, both with homogeneous and heterogeneous coefficients. We present extensive numerical results to compare the multilevel OSM, the one-level OSM, and the multigrid scheme. The experiments show that the multilevel OSM inherits robustness from the one-level OSM for heterogeneous elliptic problems, wave problems, and heterogeneous couplings. Finally, we apply the method to design a two-level solver for the heterogeneous Stokes-Darcy system
On the Derivation of Optimized Transmission Conditions for the Stokes-Darcy Coupling
No full textRecently a lot of attention has been devoted to the Stokes-Darcy coupling which is a system of equations used to model the flow of fluids in porous media. In [2, 1] a non standard behaviour of the optimized Schwarz method (OSM) has been observed: the optimized parameters obtained solving the classical min-max problems do not lead to an optimized convergence
Heterogeneous Optimized Schwarz Methods for Coupling Helmholtz and Laplace Equations
No full textOptimized Schwarz methods have increasingly drawn attention over the last decades because of their improvements in terms of robustness and computational cost compared to the classical Schwarz method. Optimized Schwarz methods are also a natural framework to study heterogeneous phenomena, where the spatial decomposition is provided by the multi-physics of the problem, because of their good convergence properties in the absence of overlap. We propose here zeroth order optimized transmission conditions for the coupling between the Helmholtz equation and the Laplace equation, giving asymptotically optimized choices for the parameters, and illustrating our analytical results with numerical experiments
Optimized Schwarz Methods for Advection Diffusion Equations in Bounded Domains
No full textOptimized Schwarz methods use better transmission conditions than the classical Dirichlet conditions that were used by Schwarz. These transmission conditions are optimized for the physical problem that needs to be solved to lead to fast convergence. The opimization is typically performed in the geometrically simplified setting of two unbounded subdomains using Fourier transforms. Recent studies for both homogeneous and heterogeneous domain decomposition methods indicate that the geometry of the physical domain has actually an influence on this opimization process. We study here this influence for an advection diffusion equation in a bounded domain using separation of variables. We provide theoretical results for the min-max problems characterizing the optimized transmission conditions. Our numerical experiments show significant improvements of the new transmission conditions which take the geometry into account, especially for strong tangential advection. </p
Analysis of the parallel Schwarz method for growing chains of fixed-sized subdomains: Part III. ETNA - Electronic Transactions on Numerical Analysis
No full textIn the ddCOSMO solvation model for the numerical simulation ofmolecules (chains of atoms), the unusual observation was made thatthe associated Schwarz domain-decomposition method converges independentlyof the number of subdomains (atoms) and this without coarsecorrection, i.e., the one-level Schwarz method is scalable.We analyzed this unusual property for the simplified caseof a rectangular molecule and square subdomains using Fourier analysis,leading to robust convergence estimates in the -norm and lateralso for chains of subdomains represented by disks using maximumprinciple arguments, leading to robust convergence estimates in. A convergence analysis in the more natural-setting proving convergence independently of the number ofsubdomains was, however, missing. We close this gap in this paperusing tools from the theory of alternating projection methodsand estimates introduced by P.-L. Lions for the study of domaindecomposition methods. We prove that robust convergenceindependently of the number of subdomains is possible also in and show furthermore that even for certain two-dimensional domainswith holes, Schwarz methods can be scalable without coarse-space corrections.As a by-product, we review some of the results of P.-L. Lions[On the Schwarz alternating method. I, in DomainDecomposition Methods for Partial Differential Equations, SIAM,Philadelphia, 1988, pp. 1-42]and in some cases provide simpler proofs
On the Scalability of Classical One-Level Domain-Decomposition Methods
No full textOne-level domain-decomposition methods are in general not scalable, and coarse corrections are needed to obtain scalability. It has however recently been observed in applications in computational chemistry that the classical one-level parallel Schwarz method is surprizingly scalable for the solution of one- and two-dimensional chains of fixed-sized subdomains. We first review some of these recent scalability results of the classical one-level parallel Schwarz method, and then prove similar results for other classical one-level domain-decomposition methods, namely the optimized Schwarz method, the Dirichlet–Neumann method, and the Neumann–Neumann method. We show that the scalability of one-level domain decomposition methods depends critically on the geometry of the domain-decomposition and the boundary conditions imposed on the original problem. We illustrate all our results also with numerical experiments
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