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Abstract multiscale-hybrid-mixed methods

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Abstract

In an abstract setting, we investigate the basic ideas behind the Multiscale Hybrid Mixed (MHM) method, a Domain Decomposition scheme designed to solve multiscale partial differential equations (PDEs) in parallel. As originally proposed, the MHM method starting point is a primal hybrid formulation, which is then manipulated to result in an efficient method that is based on local independent PDEs and a global problem that is posed on the skeleton of the finite element mesh. Recasting the MHM method in a more general framework, we investigate some conditions that yield a well-posed method. We apply the general ideas to different formulations, and, in particular, come up with an interesting and fruitful connection between the Multiscale Finite Element Method and a dual hybrid method. Finally, we propose a method that combines the main ideas of the Discontinuous Enrichment Method and the MHM method.

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Authors and Affiliations

  1. Laboratório Nacional de Computação Científica, Av. Getúlio Vargas, 333, Petrópolis-RJ, Brazil

    Alexandre L. Madureira

  2. Fundação getúlio vargas, Praia de Botafogo, 190, Rio de Janeiro-RJ, Brazil

    Alexandre L. Madureira

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  1. Alexandre L. Madureira
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Correspondence to Alexandre L. Madureira.

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Dedicated to Leo Franca, in memoriam.

We thank Frédéric Valentin for several fruitful discussions and comments that enriched the present work. We also gratefully acknowledge the partial support of CNPq, Grant numbers 308360/2010-9 and 560108/2010-9.

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Madureira, A.L. Abstract multiscale-hybrid-mixed methods. Calcolo 52, 543–557 (2015). https://doi.org/10.1007/s10092-014-0129-5

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