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Physical systems encompassing a variety of strongly coupled scales pose major computational challenges in terms of analysis modeling and simulation. In this report we first discuss a multiscale modeling approach for the transport of particles such as DNA in heterogeneous devices (microarrays). The model involves a multiscale elliptic equation coupled with a multiscale advection-diffusion equation. We introduce a numerical method for the solution of the coupled system of equations. We first discuss a new multiscale finite element method for the solution of the elliptic problem. We then explain how it is possible to use an explicit stabilized method (ROCK) for the numerical solution of the stiff system of ordinary differential equations of large dimension, originating from the method of lines discretization of the advection-diffusion equation.
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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}.