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Spectral Analysis of Dirichlet–Neumann Operators and Optimized Schwarz Methods with Robin Transmission Conditions

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Abstract

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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... In this case, the proof in Appendix A can be modified, leading (neglecting logarithmic terms) to a lower bound of (26). The resulting dependence of the convergence rate of Algorithm 1 on the ratio H/h can be observed from the numerical results in Section 4.3; furthermore, it is in agreement with the theoretical estimates of the spectrum of discrete Dirichlet-to-Neumann operators recently presented in [47]. ...
... error of Algorithm 2 with s n = 0.15 for a 2 {0.5,0.6,0.7,0.8}; left: l max = 4; right: inner iteration stopped according to(47). ...
... The first one can directly be bounded using (6.5b). For the second one, we obtain with the discrete inequalities stated in, e.g., [171], and Assumption 7.4 ...
... The resulting dependence of the convergence rate of Algorithm 1 on the ratio H/h can be observed from the numerical results in Section 7.3.3; furthermore, it is in agreeement with the theoretical estimates of the spectrum of discrete Dirichletto-Neumann operators recently presented in [171]. ...
Thesis
In many technical and engineering applications, numerical simulation is becoming more and more important for the design of products or the optimization of industrial production lines. However, the simulation of complex processes like the forming of sheet metal or the rolling of a car tire is still a very challenging task, as nonlinear elastic or elastoplastic material behaviour needs to be combined with frictional contact and dynamic effects. In addition, these processes often feature a small mobile contact zone which needs to be resolved very accurately to get a good picture of the evolution of the contact stress. In order to be able to perform an accurate simulation of such intricate systems, there is a huge demand for a robust numerical scheme that combines a suitable multiscale discretization of the geometry with an efficient solution algorithm capable of dealing with the material and contact nonlinearities. The aim of this thesis is to design such an algorithm by combining several different methods which are described in the following. Our main field of application is structural mechanics. Here, we base the implementation on finite element methods in space and implicit finite difference schemes in time. The conditions for both plasticity and frictional contact are given in terms of a set of local inequality constraints which are formulated by introducing additional inner or dual degrees of freedom. As the meshes are generally non-matching at the contact interface, we employ mortar techniques to incorporate the contact constraints in a variationally consistent way. By using biorthogonal basis functions for the discrete multiplier space, the contact conditions can be enforced node-wise, and a two-body contact problem can be solved in the same way as a one-body problem. The next step in the construction of an efficient solution algorithm is to reformulate the local inequality conditions for plasticity and contact in terms of nondifferentiable equalities. These nonlinear complementarity functions can be combined with the equations for the bulk material to form a set of nonlinear semismooth equations which are then solved by means of a generalized form of the Newton method. Due to the local structure of the inequality constraints, this iterative scheme can be implemented as an active set strategy where the active sets are updated in each Newton iteration. Further, the additional dual degrees of freedom can easily be eliminated using local static condensation. We remark that the well-known radial return method is a special case of this general framework if the plastic hardening laws are linear. However, the convergence properties of the Newton iteration strongly depend on the choice of the NCP function. In this context, we show that the function corresponding to the radial return method is not optimal, and we present a family of modified NCP functions which allow for better convergence results. Another important issue for the robust simulation of dynamic contact problems is related to the inertia terms. If standard time discretization schemes like the trapezoidal rule are used, the contact stress often shows spurious oscillations in time that become worse when the time step is refined. In order to avoid this effect, we employ a modified mass matrix where no mass is associated with the contact nodes. By this, the original semi-discrete system decouples into an algebraic equation in time for the contact nodes and an ordinary differential equation in time for the other nodes. This in turn leads to much smoother results for the contact stress. We present an efficient way of obtaining the modified mass matrix by means of non-standard quadrature formulas used only for the elements near the contact boundary. Furthermore, we prove optimal a priori error estimates for the modified semi-discrete as well as for the fully discrete system, provided that the contact stress is given and that the solution is sufficiently regular. In the last part of the thesis, we deal with the situation that the body features fine local structures near the contact zone by incorporating the multiscale aspect into the discretization. For this, the domain is decomposed into several overlapping subdomains which have different grid spacing; one global mesh that does not resolve the details and overlapping local patches with a fine triangulation. Based on a surface coupling by means of the mortar method, we construct an iterative solution scheme for the coupled problem whose convergence rate is bounded independently of the mesh size or the Lame parameters. Finally, we employ the subdomain decomposition for introducing a finer time step size on the patch. We present suitable interface conditions with no numerical dissipation and prove a priori error estimates with respect to time for the resulting coupled energy-conserving system. The latter can efficiently be solved by the iterative procedure presented before.
... In this case, the proof in Appendix A can be modified, leading (neglecting logarithmic terms) to a lower bound of (26). The resulting dependence of the convergence rate of Algorithm 1 on the ratio H/h can be observed from the numerical results in Section 4.3; furthermore, it is in agreeement with the theoretical estimates of the spectrum of discrete Dirichlet-to-Neumann operators recently presented in [47]. ...
... The active set A h n of the corresponding mortar solution is sketched on the left of Figure 8, whereas its coarse grid approximation A H n (0.15) can be seen on the right side. (47). ...
Article
Frictional dynamic contact problems with complex geometries are a challenging task – from the computational as well as from the analytical point of view – since they generally involve space and time multi-scale aspects.To be able to reduce the complexity of this kind of contact problem, we employ a non-conforming domain decomposition method in space, consisting of a coarse global mesh not resolving the local structure and an overlapping fine patch for the contact computation. This leads to several benefits: First, we resolve the details of the surface only where it is needed, i.e., in the vicinity of the actual contact zone. Second, the subproblems can be discretized independently of each other which enables us to choose a much finer time scale on the contact zone than on the coarse domain. Here, we propose a set of interface conditions that yield optimal a priori error estimates on the fine-meshed subdomain without any artificial dissipation. Further, we develop an efficient iterative solution scheme for the coupled problem that is robust with respect to jumps in the material parameters. Several complex numerical examples illustrate the performance of the new scheme.
... Let S 1 and S 2 be the standard Dirichlet-to-Neumann operators, defined in [26,30]. The functions ε n i (i = 1 or 2) restricted to interface Γ satisfies the relation ...
... It is known [30] that ...
Article
Full-text available
In this work, we solve a long-standing open problem: Is it true that the convergence rate of the Lions' Robin-Robin nonoverlapping domain decomposition(DD) method can be constant, independent of the mesh size h? We closed this twenty-year old problem with a positive answer. Our theory is also verified by numerical tests.
... It is known [1,12] that ...
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.
... 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.
... where the first transmission condition is chosen based on the fact that only the normal components of the finite element functions are continuous across the interelement boundaries, see [39,45]. The transmission conditions of (2.1) is equivalent to the following Robin-Robin boundary conditions: ...
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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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.
Chapter
An attempt is made to analyze within reasonable limits the basic mathematical aspects of the finite element method. The information given should serve as an introduction to current research on this subject. Only actual problems are covered. Theorems are given to represent important results.
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Thesis (Ph. D.)--Cornell University, May, 1989. Bibliography: leaves 187-196.
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. This paper concerns the numerical solutions of the nearly elastic wave equations with the first order absorbing boundary condition; these equations describe the motion of a nearly elastic solid in the frequency domain. Two mixed finite elements, the Johnson--Mercier element and the Arnold--Douglas--Gupta element, are adapted and analyzed for the problem. The resulting mixed finite element equations are complex--valued and are neither Hermitian nor definite. As a result, most standard iterative methods fail to converge for the systems. To solve the mixed finite element equations, a parallelizable domain decomposition iterative method is proposed. The convergence of the method is demonstrated and a rate of convergence of the form 1 Gamma Ch is derived. These results are valid for the case when the original domain is decomposed into subdomains which consist of an individual element associated with the above two mixed finite elements. x1. Introduction. Wave propagation in real media is ...
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  • Xuejun Xu
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Natural Boundary Integral Method and Its Applications The Netherlands Redistribution subject to SIAM license or copyright
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Domain decomposition method and the Helmholtz problem, in Mathematical and Numerical Aspects of Wave Propagation Phenomena
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B. Despres, Domain decomposition method and the Helmholtz problem, in Mathematical and Numerical Aspects of Wave Propagation Phenomena, G. Cohen, L. Halpern, and P. Joly, eds., SIAM, Philadelphia, 1991, pp. 44–52.