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Two-level Schwarz domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature, most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume. In this paper, we use the excellent smoothing properties of Schwarz domain decomposition methods to define, for general elliptic problems, a new class of substructured two-level methods, for which both Schwarz smoothers and coarse correction steps are defined on the interfaces (except for the application of the smoother that requires volumetric subdomain solves). This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, our approach does not require, like classical multi-grid methods, the explicit construction of coarse spaces, and it permits a multilevel extension, which is desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

Two-level domain decomposition methods are very powerful techniques for the efficient numerical solution of partial differential equations (PDEs). A two-level domain decomposition method requires two main components: a one-level preconditioner (or its corresponding smoothing iterative method), which is based on domain decomposition techniques, and a coarse correction step, which relies on a coarse space. The coarse space must properly represent the error components that the chosen one-level method is not capable to deal with. In the literature most of the works introduced efficient coarse spaces obtained as the span of functions defined on the entire space domain of the considered PDE. Therefore, the corresponding two-level preconditioners and iterative methods are defined in volume.In this paper, a new class of substructured two-level methods is introduced, for which both domain decomposition smoothers and coarse correction steps are defined on the interfaces. This approach has several advantages. On the one hand, the required computational effort is cheaper than the one required by classical volumetric two-level methods. On the other hand, it allows one to use some of the well-known efficient coarse spaces proposed in the literature. Moreover, our new substructured framework can be efficiently extended to a multi-level framework, which is always desirable when the high dimension of the problem or the scarce quality of the coarse space prevents the efficient numerical solution. Numerical experiments demonstrate the effectiveness of the proposed new numerical framework.

Due to their property of convergence in the absence of overlap, optimized Schwarz methods are the natural domain decomposition framework for heterogeneous problems, where the spatial decomposition is provided by the multiphysics of the phenomena. We study here heterogeneous problems which arise from the coupling of second order elliptic PDEs. Theoretical results and asymptotic formulas are proposed solving the corresponding min-max problems both for single and double sided optimizations, while numerical results confirm the effectiveness of our approach even when analytical conclusions are not available. Our analysis shows that optimized Schwarz methods do not suffer the heterogeneity, it is the opposite, they are faster the stronger the heterogeneity is. It is even possible to have h independent convergence choosing two independent Robin parameters. This property was proved for a Laplace equation with discontinuous coefficients, but only conjectured for more general couplings in [M. J. Gander and O. Dubois, Numer. Algorithms, 69 (2015), pp. 109--144]. Our study is completed by an application to a contaminant transport problem.

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