1,108 research outputs found

    On the initial estimate of interface forces in FETI methods

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    The Balanced Domain Decomposition (BDD) method and the Finite Element Tearing and Interconnecting (FETI) method are two commonly used non-overlapping domain decomposition methods. Due to strong theoretical and numerical similarities, these two methods are generally considered as being equivalently efficient. However, for some particular cases, such as for structures with strong heterogeneities, FETI requires a large number of iterations to compute the solution compared to BDD. In this paper, the origin of the bad efficiency of FETI in these particular cases is traced back to poor initial estimates of the interface stresses. To improve the estimation of interface forces a novel strategy for splitting interface forces between neighboring substructures is proposed. The additional computational cost incurred is not significant. This yields a new initialization for the FETI method and restores numerical efficiency which makes FETI comparable to BDD even for problems where FETI was performing poorly. Various simple test problems are presented to discuss the efficiency of the proposed strategy and to illustrate the so-obtained numerical equivalence between the BDD and FETI solvers

    Convergence and Optimality of Adaptive Mixed Finite Element Methods

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    The convergence and optimality of adaptive mixed finite element methods for the Poisson equation are established in this paper. The main difficulty for mixed finite element methods is the lack of minimization principle and thus the failure of orthogonality. A quasi-orthogonality property is proved using the fact that the error is orthogonal to the divergence free subspace, while the part of the error that is not divergence free can be bounded by the data oscillation using a discrete stability result. This discrete stability result is also used to get a localized discrete upper bound which is crucial for the proof of the optimality of the adaptive approximation

    Goal-oriented h-adaptivity for the Helmholtz equation: error estimates, local indicators and refinement strategies

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    The final publication is available at Springer via http://dx.doi.org/10.1007/s00466-010-0557-2This paper introduces a new goal-oriented adaptive technique based on a simple and effective post-process of the finite element approximations. The goal-oriented character of the estimate is achieved by analyzing both the direct problem and an auxiliary problem, denoted as adjoint or dual problem, which is related to the quantity of interest. Thus, the error estimation technique proposed in this paper would fall into the category of recovery-type explicit residual a posteriori error estimates. The procedure is valid for general linear quantities of interest and it is also extended to non-linear ones. The numerical examples demonstrate the efficiency of the proposed approach and discuss: (1) different error representations, (2) assessment of the dispersion error, and (3) different remeshing criteria.Peer ReviewedPostprint (author's final draft

    Macroscopic network circulation for planar graphs

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    The analysis of networks, aimed at suitably defined functionality, often focuses on partitions into subnetworks that capture desired features. Chief among the relevant concepts is a 2-partition, that underlies the classical Cheeger inequality, and highlights a constriction (bottleneck) that limits accessibility between the respective parts of the network. In a similar spirit, the purpose of the present work is to introduce a new concept of maximal global circulation and to explore 3-partitions that expose this type of macroscopic feature of networks. Herein, graph circulation is motivated by transportation networks and probabilistic flows (Markov chains) on graphs. Our goal is to quantify the large-scale imbalance of network flows and delineate key parts that mediate such global features. While we introduce and propose these notions in a general setting, in this paper, we only work out the case of planar graphs. We explain that a scalar potential can be identified to encapsulate the concept of circulation, quite similarly as in the case of the curl of planar vector fields. Beyond planar graphs, in the general case, the problem to determine global circulation remains at present a combinatorial problem
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