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In this paper we discuss partial differential equations with multiple scales for which scale resolution are needed in some subregions, while a separation of scale and numerical homogenization is possible in the remaining part of the computational domain. Departing from the classical coupling approach that often relies on artificial boundary conditions computed from some coarse grain simulation, we propose a coupling procedure in which virtual boundary conditions are obtained from the minimization of a coarse grain and a fine scale model in overlapping regions where both models are valid. We discuss this method with a focus on interface control and a numerical strategy based on non-matching meshes in the overlap. A fully discrete a priori error analysis of the heterogeneous coupled multiscale method is derived and numerical experiments that illustrate the efficiency and flexibility of the proposed strategy are presented.
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how?' but also why?', where?' and what for?'.
The motivation for developing structure-preserving algorithms for special classes of problems originates independently in such diverse areas of research as astronomy, molecular dynamics, mechanics, control theory, theoretical physics and numerical analysis, with important contributions from other areas of both applied and pure mathematics. Moreover, it turns out that preservation of geometric properties of the flow not only produces an improved qualitative behaviour, but also allows for a significantly more accurate long-time integration than with general-purpose methods.
In addition to the construction of geometric integrators, an important aspect of geometric integration is the light it sheds on the relationship between geometric properties of a numerical method and favourable error propagation in long-time integration. A very successful organising principle is backward error analysis, whereby the numerical one-step map is interpreted as (almost) the flow of a modified differential equation. In this way, geometric properties of the numerical integrator translate seamlessly into structure preservation on the level of the modified equation. Much insight and rigourous error estimates over long time intervals can then be obtained by combining backward error analysis with the KAM theory and related perturbation theories for Hamiltonian and reversible systems. While this approach has been very successful for ordinary differential equations, much less is currently known about highly oscillatory systems and geometric integrators for partial differential equations.
Geometric numerical integration has been an active interdisciplinary research area since the last decade. Although the subject is in a lively phase of intensive development, the results so far are substantive and they impact on a wide range of application areas and on our understanding of core issues in computational mathematics. This is evidenced by the monographs \cite{HLW:GNI2002,LR:SMH2004}.