Simulation and analysis of temporal multiscale problems with partial differential equations

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This research project aims at the analysis of temporal multiscale problems involving partial differential equations. Many applications describe long-term effects, such as material weathering, material fracture due to atomistic defects such as impurities, or biological pattern formation and growth processes. These phenomena are often influenced by important short-scale effects.Simulation of such processes with traditional techniques is not possible. Formation of arteriosclerotic plaques is a slow process that takes months. It is however strongly influenced by the pulsating blood flow which will require a resolution of less than a second. Direct numerical simulations of complex flow problems with such a fine resolution over long periods of time are clearly beyond the bounds of possibility. We will develop and analyze multiscale methods in time, that are based on averaging the fast processes, such that effective long term problems can be considered.One part of this project is devoted to the mathematical analysis of temporal multiscale problems. Usually, we can introduce a scale parameter that indicates the relation between fast and slow scales. We will investigate the convergence of the solution to the temporal multiscale problem to the solution of the simpler averaged long-term problem. Convergence will be measured with respect to the scale parameter.In the second part of this project, we will design and implement numerical approximation schemes for the efficient simulation of temporal multiscale problems. These numerical tools will aim at approximating the solution to the averaged long-term problem. The numerical methods will be based on finite elements for spatial discretization of the partial differential equations and Galerkin methods for temporal discretization. For deriving efficient simulation tools, we will base the discretizations on adaptivity in space in time. Both parts are conducted in joint effort. For designing numerical approximation tools, we must know about the analytical properties of the involved equations, such that correct averaging schemes can be designed. Numerical experiments will help the analysis by providing first impressions on expected convergence rates. We develop numerical schemes of a universal character without a limitation to specific applications. All methods will be implemented in the finite element software library Gascoigne and published as open source project, such that new findings are available for various related multiscale problems. The mathematical investigation of temporal multiscale problems with partial differential equations is a challenging task. Up to now, only very few results are available.