On substructuring algorithms and solution techniques for the numerical approximation of partial differential equations

Gunzburger, Max; Nicolaides, R. A.

doi:10.1016/0168-9274(86)90031-0

Cited by 7 publications

(1 citation statement)

References 5 publications

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“…These methods have a long history in the mathematical and engineering literatures; see, e.g., [1,30]. It is shown in [17] that the optimization-based approach employed here recovers many known substructuring methods and also defines some new methods.…”

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confidence: 99%

A Gradient Method Approach to Optimization-Based Multidisciplinary Simulations and Nonoverlapping Domain Decomposition Algorithms

Du¹,

Gunzburger²

2000

SIAM J. Numer. Anal.

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Abstract. It has been shown recently that optimization-based nonoverlapping domain decomposition algorithms are connected to many well-known algorithms. Using a gradient-type iterative strategy for the optimization problem, we present further discussion on how to develop various algorithms that can integrate subdomain solvers into a solver for the problem in the whole domain. In particular, the algorithms we discuss can be used to develop efficient solvers of multidisciplinary problems which are constructed using existing subdomain solvers without the need for making changes in the latter.Key words. multidisciplinary simulations and optimization, domain decomposition, nonoverlapping subdomains, optimization, convergence, finite element, parallel computation AMS subject classifications. 65N55, 65N30, 65Y10, 35J20, 65K10PII. S00361429983430871. Introduction. Domain decomposition is a realization of the divide and conquer strategy in which one attempts to treat a (computationally) intractable problem by replacing it by two or more simpler problems, each of which is presumably tractable. In particular, domain decomposition methodologies provide an effective means of introducing parallelism into problems which may not exhibit obvious, inherent parallelism.The decomposition of problems based on a division of the computational domain into two or more subdomains is useful in two contexts. First, one may introduce (often artificial) subdomains and then define problems over the subdomains which, when solved repeatedly through an iterative procedure, yield practically the same solution as that of the original problem. Such algorithms have been studied extensively in the past decade; see, e.g., [1,3,6,7,8,9,15,20,21,23,32,33]. On the other hand, a domain decomposition methodology can also be viewed as an integrator of subproblem solvers in a complicated system; see, e.g., [24,38,39]. Specifically, we have in mind problems where the subdivision into subdomains occurs naturally, resulting from changes in the mathematical models from one subdomain to another. In the simplest realization, we merely have different data in the model equations in the different subdomains. For example, we could have a wave propagation problem or a heat transfer problem wherein the media properties, e.g., the speed of sound or the diffusion coefficient, respectively, are piecewise continuous. In more complicated settings, we have completely different model equations in the different subdomains. For example, for aeroelastic problems, fluid dynamics equations in a flow

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