Simulation and analysis of temporal multiscale problems with partial differential equations

本研究项目旨在分析涉及偏微分方程组的时态多尺度问题。许多应用描述了长期效应，如材料风化、由于原子缺陷(如杂质)导致的材料断裂，或生物图案的形成和生长过程。这些现象往往受到重要的短尺度效应的影响，不可能用传统的技术来模拟这些过程。动脉硬化斑块的形成是一个缓慢的过程，需要数月时间。然而，它受到脉动血流的强烈影响，这将需要不到一秒的分辨率。在很长一段时间内，以如此精细的分辨率对复杂流动问题进行直接数值模拟显然是不可能的。我们将及时开发和分析基于平均快速过程的多尺度方法，以便能够考虑有效的长期问题。本项目的一部分致力于时态多尺度问题的数学分析。通常，我们可以引入一个标度参数来表示快尺度和慢尺度之间的关系。我们将研究时间多尺度问题的解与更简单的平均长期问题的解的收敛。在这个项目的第二部分，我们将设计和实现数值逼近方案来有效地模拟时间多尺度问题。这些数字工具将旨在近似解决平均长期问题的解决方案。数值方法将基于有限元对偏微分方程组进行空间离散，伽辽金方法对时间进行离散。为了得到高效的仿真工具，我们将基于空间和时间的自适应性进行离散化。这两部分都是共同努力进行的。在设计数值逼近工具时，我们必须了解所涉及方程的分析性质，以便能够设计正确的平均方案。数值实验将通过提供对预期收敛速度的第一印象来帮助分析。我们开发了不受特定应用限制的具有普遍特征的数值格式。所有方法都将在有限元软件库Gascoigne中实现，并作为开源项目发布，以便对各种相关的多尺度问题提供新的发现。时态多尺度偏微分方程组问题的数学研究是一项具有挑战性的任务。到目前为止，可用的结果很少。

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.