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A Robin-type Domain Decomposition Method with Red-Black Partition

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Abstract

In this paper, we present a new Robin-type nonoverlapping domain decomposition (DD) preconditioner. The unknown variables to be solved in this preconditioned algebraic system are the Robin transmission data on the interface, which are different from the well-known DD methods like substructuring nonoverlapping DD method and FETI method. Through choosing suitable parameter on each subdomain boundary and using the tool of energy estimate, for the second-order elliptic problem, we prove that the condition number of the preconditioned system is C(1 + log(H/h))(2), where H is the coarse mesh size and h is the fine mesh size. Moreover, in this paper, we always assume that there is a red-black partition for the whole domain Omega. Numerical results are given to illustrate the efficiency of our DD preconditioner.

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... For any g ∈ W Γ , it holds that [5] ...
... where δ > 0 is an constant independent of h, H. For the details, we refer to [5]. By (4.6) and (4.7), we may get the upper bound estimate, i.e. ...
Preprint
In this paper, we revisit the nonoverlapping domain decomposition methods for solving elliptic problems with high contrast coefficients. Some interesting results are discovered. We find that the Dirichlet-Neumann algorithm and Robin-Robin algorithms may make full use of the ratio of coefficients. Actually, in the case of two subdomains, we show that their convergence rates are O(ϵ)O(\epsilon), if ν1ν2\nu_1\ll\nu_2, where ϵ=ν1/ν2\epsilon = \nu_1/\nu_2 and ν1,ν2\nu_1,\nu_2 are coefficients of two subdomains. Moreover, in the case of many subdomains, the condition number bounds of Dirichlet-Neumann algorithm and Robin-Robin algorithm are 1+ϵ(1+log(H/h))21+\epsilon(1+\log(H/h))^2 and C+ϵ(1+log(H/h))2C+\epsilon(1+\log(H/h))^2, respectively, where ϵ\epsilon may be a very small number in the high contrast coefficients case. Besides, the convergence behaviours of the Neumann-Neumann algorithm and Dirichlet-Dirichlet algorithm may be independent of coefficients while they could not benefit from the discontinuous coefficients. Numerical experiments are preformed to confirm our theoretical findings.
... in [16,25]. Then, in [29], Gander, Halpern, and Nataf improved it to be [30], and further they used the Dirichlet-to-Neumann operator to prove that the convergence rate cannot be improved. A sharp convergent result is also obtained in [31][32][33][34][35]. ...
... In [30,31], the authors introduced some preconditioned systems of the optimized Schwarz method. In [29], the authors present a new Robin-type nonoverlapping domain decomposition (DD) preconditioner, and they have proved the preconditioner is optimal and scalable. ...
... Domain decomposition (DD) methods are powerful parallel methods for solving the systems arising from finite element discretization of elliptic problems. There exist many well known nonoverlapping DD methods for solving indefinite systems of Helmholtz equations, like the Robin-type DD method [17,18], the substructuring method [19], the finite element tearing and interconnecting (FETI) method [20,21] and the dual-primal finite element tearing and interconnecting (FETI-DP) method [22,23]. Alternative advanced nonoverlapping DD method is the balancing domain decomposition by constraints (BDDC) methods [24,25,26]. ...
Preprint
Balancing domain decomposition by constraints (BDDC) algorithms with adaptive primal constraints are developed in a concise variational framework for the weighted plane wave least-squares (PWLS) discritization of Helmholtz equations with high and various wave numbers. The unknowns to be solved in this preconditioned system are defined on elements rather than vertices or edges, which are different from the well-known discritizations such as the classical finite element method. Through choosing suitable "interface" and appropriate primal constraints with complex coefficients and introducing some local techniques, we developed a two-level adaptive BDDC algorithm for the PWLS discretization, and the condition number of the preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is only dependent on the maximum number of interfaces per subdomain. A multilevel algorithm is also attempted to resolve the bottleneck in large scale coarse problem. Numerical results are carried out to confirm the theoretical results and illustrate the efficiency of the proposed algorithms.
... It seems possible only at the discrete level to have a convergence rate of the non-overlapping optimized Schwarz method. Qin and Xu (2006) got the first estimate of the convergence factor 1 − O(h 1/2 H −1/2 ) with an optimized choice of the Robin parameter; see also Qin, Shi and Xu (2008), Xu and Qin (2010), Lui (2009), Loisel (2013, Liu and Xu (2014), Gander and Hajian (2015), Gander and Hajian (2018). In the overlapping case, the literature becomes even sparser, and there is only the work of Loisel and Szyld (2010) to our knowledge. ...
Preprint
Full-text available
Schwarz methods use a decomposition of the computational domain into subdomains and need to put boundary conditions on the subdomain boundaries. In domain truncation one restricts the unbounded domain to a bounded computational domain and also needs to put boundary conditions on the computational domain boundaries. It turns out to be fruitful to think of the domain decomposition in Schwarz methods as truncation of the domain onto subdomains. The first truly optimal Schwarz method that converges in a finite number of steps was proposed in 1994 and used precisely transparent boundary conditions as transmission conditions between subdomains. Approximating these transparent boundary conditions for fast convergence of Schwarz methods led to the development of optimized Schwarz methods -- a name that has become common for Schwarz methods based on domain truncation. Compared to classical Schwarz methods which use simple Dirichlet transmission conditions and have been successfully used in a wide range of applications, optimized Schwarz methods are much less well understood, mainly due to their more sophisticated transmission conditions. This present situation is the motivation for our survey: to give a comprehensive review and precise exploration of convergence behaviors of optimized Schwarz methods based on Fourier analysis taking into account the original boundary conditions, many subdomain decompositions and layered media. The transmission conditions we study include the lowest order absorbing conditions (Robin), and also more advanced perfectly matched layers (PML), both developed first for domain truncation.
... Domain decomposition (DD) methods are powerful parallel methods for solving the systems arising from finite element discretization of elliptic problems. There exist many well known nonoverlapping DD methods for solving indefinite systems of Helmholtz equations, like the Robin-type DD method [17,18], the substructuring method [19], the finite element tearing and interconnecting (FETI) method [20,21] and the dual-primal finite element tearing and interconnecting (FETI-DP) method [22,23]. Alternative advanced nonoverlapping DD method is the balancing domain decomposition by constraints (BDDC) methods [24,25,26]. ...
Article
Balancing domain decomposition by constraints (BDDC) algorithms with adaptive primal constraints are developed in a concise variational framework for the weighted plane wave least-squares (PWLS) discritization of Helmholtz equations with high and various wave numbers. The unknowns to be solved in this preconditioned system are defined on elements rather than vertices or edges, which are different from the well-known discritizations such as the classical finite element method. Through choosing suitable "interface" and appropriate primal constraints with complex coefficients and introducing some local techniques, we developed a two-level adaptive BDDC algorithm for the PWLS discretization, and the condition number of the preconditioned system is proved to be bounded above by a user-defined tolerance and a constant which is only dependent on the maximum number of interfaces per subdomain. A multilevel algorithm is also attempted to resolve the bottleneck in large scale coarse problem. Numerical results are carried out to confirm the theoretical results and illustrate the efficiency of the proposed algorithms.
Article
Schwarz methods use a decomposition of the computational domain into subdomains and need to impose boundary conditions on the subdomain boundaries. In domain truncation one restricts the unbounded domain to a bounded computational domain and must also put boundary conditions on the computational domain boundaries. In both fields there are vast bodies of literature and research is very active and ongoing. It turns out to be fruitful to think of the domain decomposition in Schwarz methods as a truncation of the domain onto subdomains. Seminal precursors of this fundamental idea are papers by Hagstrom, Tewarson and Jazcilevich (1988), Després (1990) and Lions (1990). The first truly optimal Schwarz method that converges in a finite number of steps was proposed by Nataf (1993), and used precisely transparent boundary conditions as transmission conditions between subdomains. Approximating these transparent boundary conditions for fast convergence of Schwarz methods led to the development of optimized Schwarz methods – a name that has become common for Schwarz methods based on domain truncation. Compared to classical Schwarz methods, which use simple Dirichlet transmission conditions and have been successfully used in a wide range of applications, optimized Schwarz methods are much less well understood, mainly due to their more sophisticated transmission conditions. A key application of Schwarz methods with such sophisticated transmission conditions turned out to be time-harmonic wave propagation problems, because classical Schwarz methods simply do not work in this case. The past decade has given us many new Schwarz methods based on domain truncation. One review from an algorithmic perspective (Gander and Zhang 2019) showed the equivalence of many of these new methods to optimized Schwarz methods. The analysis of optimized Schwarz methods, however, is lagging behind their algorithmic development. The general abstract Schwarz framework cannot be used for the analysis of these methods, and thus there are many open theoretical questions about their convergence. Just as for practical multigrid methods, Fourier analysis has been instrumental for understanding the convergence of optimized Schwarz methods and for tuning their transmission conditions. Similar to local Fourier mode analysis in multigrid, the unbounded two-subdomain case is used as a model for Fourier analysis of optimized Schwarz methods due to its simplicity. Many aspects of the actual situation, e.g. boundary conditions of the original problem and the number of subdomains, were thus neglected in the unbounded two-subdomain analysis. While this gave important insight, new phenomena beyond the unbounded two-subdomain models were discovered. This present situation is the motivation for our survey: to give a comprehensive review and precise exploration of convergence behaviours of optimized Schwarz methods based on Fourier analysis, taking into account the original boundary conditions, many-subdomain decompositions and layered media. We consider as our model problem the operator Δ+η-\Delta + \eta in the diffusive case η>0\eta>0 (screened Laplace equation) or the oscillatory case η<0\eta <0 (Helmholtz equation), in order to show the fundamental difference in behaviour of Schwarz solvers for these problems. The transmission conditions we study include the lowest-order absorbing conditions (Robin), and also more advanced perfectly matched layers (PMLs), both developed first for domain truncation. Our intensive work over the last two years on this review has led to several new results presented here for the first time: in the bounded two-subdomain analysis for the Helmholtz equation, we see strong influence of the original boundary conditions imposed on the global problem on the convergence factor of the Schwarz methods, and the asymptotic convergence factors with small overlap can differ from the unbounded two-subdomain analysis. In the many-subdomain analysis, we find the scaling with the number of subdomains, e.g. when the subdomain size is fixed, robust convergence of the double-sweep Schwarz method for the free-space wave problem, either with fixed overlap and zeroth-order Taylor conditions or with a logarithmically growing PML, and we find that Schwarz methods with PMLs work like smoothers that converge faster for higher Fourier frequencies; in particular, for the free-space wave problem, plane waves (in the error) passing through interfaces at a right angle converge more slowly. In addition to our main focus on analysis in Sections 2 and 3, we start in Section 1 with an expository historical introduction to Schwarz methods, and in Section 4 we give a brief interpretation of the recently proposed optimal Schwarz methods for decompositions with cross-points from the viewpoint of transmission conditions. We conclude in Section 5 with a summary of open research problems. In Appendix A we provide a Matlab program for a block LU form of an optimal Schwarz method with cross-points, and in Appendix B we give the Maple program for the two-subdomain Fourier analysis.
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In this paper, we develop a nonoverlapping domain decomposition method for Stokes equations by mixed finite elements with discontinuous pressures. Both conforming and nonconforming finite element spaces are considered for velocities. With Robin boundary condition set on the interface, the indefinite Stokes problem is reduced to a positive definite problem for the interface Robin transmission data by a Schur complement procedure. Choosing an appropriate relaxation parameter and two parameters in the Robin boundary conditions, the algorithm may be proved optimal. Based on the Robin-type domain decomposition method, a new preconditioner for the Stokes problem is proposed. Numerical results are given to support our theoretical findings.
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In this paper, I present a Robin-type nonoverlapping domain decomposition (DD) preconditioner for the well-posed Maxwell's equations in two dimensions, which is deduced from the renew equation of the Robin-Robin iteration algorithm. The unknown variables to be solved in this preconditioned algebraic system are the Robin transmission data on the interface. Through choosing suitable parameter on each subdomain boundary and using the tool of energy estimate, I prove that the condition number of the preconditioned system is optimal. Numerical results which confirm my theory are given in the last.
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Domain decomposition methods provide powerful preconditioners for the iterative solution of the large systems of algebraic equations that arise in finite element or finite difference approximations of partial differential equations. The preconditioners are constructed from exact or approximate solvers for the same partial differential equation restricted to a set of subregions into which the given region has been divided. In addition, the preconditioner is often augmented by a coarse, second-level approximation that provides additional, global exchange of information that can enhance the rate of convergence considerably. The iterative substructuring methods, based on decompositions of the region into nonoverlapping subregions, form one of the main families of such algorithms. Many domain decomposition algorithms can conveniently be described and analyzed as Schwarz methods. These algorithms are fully defined in terms of a set of subspaces and auxiliary bilinear forms. A general theoretical framework has previously been developed. In this paper, these techniques are used in an analysis of iterative substructuring methods for elliptic problems in three dimensions. A special emphasis is placed on the difficult problem of designing good coarse models and obtaining robust methods for which the rate of convergence is insensitive to large variations in the coefficients of the differential equation. Domain decomposition algorithms can conveniently be built from modules that represent local and global components of the preconditioner. In this paper, a number of such possibilities are explored, and it is demonstrated how a great variety of fast algorithms can be designed and analyzed.
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In contrast to the positive definite Helmholtz equation, the deceivingly similar looking indefinite Helmholtz equation is difficult to solve using classical iterative methods. Simply using a Krylov method is much less effective, especially when the wave number in the Helmholtz operator becomes large, and also algebraic preconditioners such as incomplete LU factorizations do not remedy the situation. Even more powerful preconditioners such as classical domain decomposition and multigrid methods fail to lead to a convergent method, and often behave differently from their usual behavior for positive definite problems. For example increasing the overlap in a classical Schwarz method degrades its performance, as does increasing the number of smoothing steps in multigrid. The purpose of this review paper is to explain why classical iterative methods fail to be effective for Helmholtz problems, and to show different avenues that have been taken to address this difficulty.
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Optimized Schwarz methods form a class of domain decomposition methods for the solution of elliptic partial differential equations. When the subdomains are overlapping or nonoverlapping, these methods employ the optimal value of parameter(s) in the boundary condition along the artificial interface to accelerate its convergence. In the literature, the analysis of optimized Schwarz methods rely mostly on Fourier analysis and so the domains are restricted to be regular (rectangular). As in earlier papers, the interface operator can be expressed in terms of Poincaré–Steklov operators. This enables the derivation of an upper bound for the spectral radius of the interface operator on essentially arbitrary geometry. The problem of interest here is a PDE with a discontinuous coefficient across the artificial interface. We derive convergence estimates when the mesh size h along the interface is small and the jump in the coefficient may be large. We consider two different types of Robin transmission conditions in the Schwarz iteration: the first one leads to the best estimate when h is small, whereas for the second type, we derive a convergence estimate inversely proportional to the jump in the coefficient. This latter result improves upon the rate of popular domain decomposition methods such as the Neumann–Neumann method or FETI-DP methods, which was shown to be independent of the jump.
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A one-parameter generalization of P. L. Lions’ nonoverlapping domain decomposition method [Proc. 3rd Int. Symp. Houston/TX (USA) 1989, 202–223 (1990; Zbl 0704.65090)] for linear elliptic partial differential equations (PDEs) is proposed and studied. The generalized methods are shown to be descent-direction methods for minimizing an interface bias functional. Iteration convergence of both the continuous and finite element versions of the proposed methods is established. It is theoretically and numerically demonstrated that for generic choices of the parameter the generalized methods converge faster than Lions’ original method. Algorithms are given and numerical results are presented.
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In recent years, a nonoverlapping Robin-type domain decomposition method (DDM) for the finite element discretization systems of second order elliptic equations, which is based on using Robin-type boundary conditions as information transmission conditions on the subdomain interfaces, has been developed and analyzed since it was first proposed by P. L. Lions [Proceedings of the 3rd International Symposium on Domain Decomposition Methods for Partial Differential Equations, SIAM, Philadelphia, PA, 1989, 202–223 (1990; Zbl 0704.65090)]. However, the convergence rate of this DDM with many subdomains remains open when the lower term of the equations vanishes. This open problem will be considered in this paper. The convergence rate is almost 1-O(h 1/2 H -1/2 ) in certain cases-for example, the case of a small number of subdomains, where h is the mesh size and H is the size of subdomain. In order to get the desired convergence results, two mathematical skills are introduced in this paper; one is complexification of real linear spaces and the other is the spectral radius formula.
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In this paper, the tight relationship between Dirichlet—Neumann (D-N) operators and optimized Schwarz methods with Robin transmission conditions is disclosed. We describe the spectral distribution of continuous D-N operators and give a rigorous spectral analysis of discrete D-N operators. By these results, we prove that the optimized Schwarz methods with Robin transmission conditions cannot converge geometrically in the case of continuous problems. Furthermore, we get the accurate convergence rate of the two-subdomain case. In addition, an estimation of convergence rate of the optimized Schwarz methods is presented in the general case. Most of our results are asymptotically sharp.
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A preconditioner for substructuring based on constrained energy minimization concepts is presented. The preconditioner is applicable to both structured and unstructured meshes and offers a straightforward approach for the iterative solution of second- and fourth-order structural mechanics problems. The approach involves constraints associated with disjoint sets of nodes on substructure boundaries. These constraints provide the means for preconditioning at both the substructure and global levels. Numerical examples are presented that demonstrate the good performance of the method in terms of iterations, compute time, and condition numbers of the preconditioned equations.
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Several domain decomposition methods of Neumann-Neumann type are considered for solving the large linear systems of algebraic equations that arise from discretizations of elliptic problems by finite elements. We will only consider problems in three dimensions. Several new variants of the basic algorithm are introduced in a Schwarz method framework that provides tools which have already proven very useful in the design and analysis of other domain decomposition and multi-level methods. The Neumann-Neumann algorithms have several advantages over other domain decomposition methods. The subregions, which define the subproblems, only share the boundary degrees of freedom with their neighbors. The subregions can also be of quite arbitrary shape and many of the major components of the preconditioner can be constructed from subprograms available in standard finite element program libraries. In its original form, however, the algorithm lacks a mechanism for global transportation of information and its performance therefore suffers when the number of subregions increases. In the new variants of the algorithms, considered in this paper, the preconditioners include global components, of low rank, to overcome this difficulty. Bounds are established for the condition number of the iteration operator, which are independent of the number of subregions, and depend only polylogarithmically on the number of degrees of freedom of individual local subproblems. Results are also given for problems with arbitrarily large jumps in the coefficients across the interfaces separating the subregions.
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The FETI and Neumann-Neumann families of algorithms are among the best known and most severely tested domain decomposition methods for elliptic partial differential equations. They are iterative substructuring methods and have many algorithmic components in common but there are also differences. The purpose of this paper is to further unify the theory for these two families of methods and to introduce a new family of FETI algorithms. Bounds on the rate of convergence, which are uniform with respect to the coefficients of a family of elliptic problems with heterogeneous coefficients, are established for these new algorithms. The theory for a variant of the Neumann-Neumann algorithm is also redeveloped stressing similarities to that for the FETI methods.
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In this paper, certain iterative substructuring methods with Lagrange multipliers are considered for elliptic problems in three dimensions. The algorithms belong to the family of dual-primal FETI methods which have recently been introduced and analyzed successfully for elliptic problems in the plane. The family of algorithms for three dimensions is extended and a full analysis is provided for the new algorithms. Particular attention is paid to finding algorithms with a small primal subspace since that subspace represents the only global part of the dual-primal preconditioner. It is shown that the condition numbers of several of the dual-primal FETI methods can be bounded polylogarithmically as a function of the dimension of the individual subregion problems and that the bounds are otherwise independent of the number of subdomains, the mesh size, and jumps in the coefficients. These results closely parallel those for other successful iterative substructuring methods of primal as well as dual type.