Numerical experiments on a domain decomposition algorithm for nonlinear elliptic boundary value problems

Hagstrom, Thomas; Tewarson, R. P.; Jazcilevich, Arón

doi:10.1016/0893-9659(88)90097-3

Cited by 46 publications

(30 citation statements)

References 3 publications

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“…In a short note on nonlinear problems [24], Hagstrom, Tewarson, and Jazcilevich introduced Robin transmission conditions between subdomains and suggested, "Indeed, we advocate the use of nonlocal conditions." Later and independently, Tang introduced in [39] the generalized Schwarz alternating method, which uses a weighted average of Dirichlet and Neumann conditions at the interfaces, which is equivalent to a Robin condition.…”

Section: Introductionmentioning

confidence: 99%

Optimized Schwarz Methods

Gander¹

2006

SIAM J. Numer. Anal.

369

460

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Abstract. Optimized Schwarz methods are a new class of Schwarz methods with greatly enhanced convergence properties. They converge uniformly faster than classical Schwarz methods and their convergence rates dare asymptotically much better than the convergence rates of classical Schwarz methods if the overlap is of the order of the mesh parameter, which is often the case in practical applications. They achieve this performance by using new transmission conditions between subdomains which greatly enhance the information exchange between subdomains and are motivated by the physics of the underlying problem. We analyze in this paper these new methods for symmetric positive definite problems and show their relation to other modern domain decomposition methods like the new Finite Element Tearing and Interconnect (FETI) variants. There were three main motivations: the first motivation for different transmission conditions came from the nonoverlapping variant of the Schwarz method proposed by Lions, since without overlap the classical Schwarz method does not converge. Lions proposed to use Robin conditions to obtain a convergent algorithm in [31]. At the end in his paper, we find the following remark: "First of all, it is possible to replace the constants in the Robin conditions by two proportional functions on the interface, or even by local or nonlocal operators." Lions then gives a simple example in one dimension and shows that the optimal choice for the parameters in the Robin transmission conditions of the algorithm are constants in that case. In higher dimensions, however, the optimal choice involves a nonlocal transmission operator, as was shown for a two-dimensional convection diffusion problem by Charton, Nataf, and Rogier in [5], where a parabolic factorization of the operator was used to derive the optimal transmission conditions. Since nonlocal operators are not convenient to implement and costly ("ils se prêtent peu au calcul numérique" [5]), the authors propose for the convection dominated convection-diffusion problem to expand the symbols of the nonlocal operators in the small viscosity parameter to obtain local approximations. A different approximation using a Taylor expansion in the frequency parameter to obtain local transmission conditions for the convection diffusion equation is proposed in [33]; see also [34] and [32]. For symmetric coercive problems, a formulation of the nonoverlapping Schwarz method with Robin transmission conditions which avoids the explicit use of normal derivatives was introduced independently in [7], and convergence of the resulting algorithm was proved using energy estimates. A first optimization of the

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Section: Introductionmentioning

confidence: 99%

Optimized Schwarz Methods

Gander¹

2006

SIAM J. Numer. Anal.

369

460

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“…The third motivation was that the convergence rate of the classical Schwarz method is rather slow and too strongly dependent on the size of the overlap. In a short note on nonlinear problems [26], Hagstrom, Tewarson, and Jazcilevich introduced Robin transmission conditions between subdomains and suggested nonlocal operators for the best performance. In [4], these optimal, nonlocal transmission conditions were developed for advection-diffusion problems, with local approximations for small viscosity, and low order frequency approximations were proposed in [29,9].…”

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confidence: 99%

Optimized Schwarz Methods for Maxwell's Equations

Dolean¹,

Gander²,

Gerardo-Giorda³

2009

SIAM J. Sci. Comput.

137

117

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Abstract. Over the last two decades, classical Schwarz methods have been extended to systems of hyperbolic partial differential equations, using characteristic transmission conditions, and it has been observed that the classical Schwarz method can be convergent even without overlap in certain cases. This is in strong contrast to the behavior of classical Schwarz methods applied to elliptic problems, for which overlap is essential for convergence. More recently, optimized Schwarz methods have been developed for elliptic partial differential equations. These methods use more effective transmission conditions between subdomains than the classical Dirichlet conditions, and optimized Schwarz methods can be used both with and without overlap for elliptic problems. We show here why the classical Schwarz method applied to both the time harmonic and time discretized Maxwell's equations converges without overlap: the method has the same convergence factor as a simple optimized Schwarz method for a scalar elliptic equation. Based on this insight, we develop an entire new hierarchy of optimized overlapping and nonoverlapping Schwarz methods for Maxwell's equations with greatly enhanced performance compared to the classical Schwarz method. We also derive for each algorithm asymptotic formulas for the optimized transmission conditions, which can easily be used in implementations of the algorithms for problems with variable coefficients. We illustrate our findings with numerical experiments. 1. Introduction. Schwarz algorithms experienced a second youth over the last decades when distributed computers became more and more powerful and available. Fundamental convergence results for the classical Schwarz methods were derived for many partial differential equations and can now be found in several authoritative reviews [3,41,42] and books [34,33,39]. The Schwarz methods were also extended to systems of partial differential equations, such as the time harmonic Maxwell's equations [12,8], the time discretized Maxwell's equations [38], or to linear elasticity [18,19], but much less is known about the behavior of the Schwarz methods applied to hyperbolic systems of equations. This is true, in particular, for the Euler equations, to which the Schwarz algorithm was first applied in [31,32], where classical (characteristic) transmission conditions are used at the interfaces, or with more general transmission conditions in [7]. The analysis of such algorithms applied to systems proved to be very different from the scalar case; see [14,15].Over the last decade, a new class of overlapping Schwarz methods was developed for scalar partial differential equations, namely, the optimized Schwarz methods. These methods are based on a classical overlapping domain decomposition, but they use more effective transmission conditions than the classical Dirichlet conditions at the interfaces between subdomains. New transmission conditions were originally proposed for three different reasons: first, to obtain Schwarz algorithms that are convergent without...

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“…But Figure 5 suggests that one could use the overlap for the high-frequency modes, and the transmission condition for the low-frequency modes, in order to obtain a method effective for all modes in a Helmholtz problem. In addition, it might be possible to choose an even better transmission condition, as indicated toward the end in Lions' work [51], and also by Hagström et al in [42]. All these observations and further developments led at the turn of the century to the invention of the new class of optimized Schwarz methods [33], with specialized variants for Helmholtz problems [34,32].…”

Section: Domain Decomposition Methods For Helmholtz Problemsmentioning

confidence: 99%