Abstract Multiscale–hybrid–mixed Methods

MULTISCALE–HYBRID–MIXED METHODS ALEXANDRE L. MADUREIRA Dedicated to Leo Franca, in memoriam. 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. 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. 1. THE ORIGINAL PROBLEM Multiscale problems are, by definition, difficult to solve computationally, and numerical methods with domain decomposition flavor are appealing, since they woo parallel implementations. Some important ideas have been around for a while, for instance in the seminal paper [20]. More recent attempts to solve multiscale PDEs using domain decomposition or hybrid formulations include [2, 3, 15, 19]. Several theoretical aspects and applications involving hybrid methods appear, for instance, in [4, 11–13, 26], and references therein. An interesting proposal by Harder, Paredes and Valentin [23] came out also recently. Aiming to solve heterogeneous Darcy equations using a primal hybrid method, the authors came up with an efficient formulation that is well-posed and easy to implement in parallel. They christened the scheme Multiscale Hybrid Mixed, or MHM for short. Date: September 25, 2014.