Simulation and analysis of temporal multiscale problems with partial differential equations

偏微分方程时态多尺度问题的模拟与分析

• 批准号: 411046898
• 负责人: Professor Dr. Thomas Richter
• 金额: --
• 依托单位: Institut für Analysis und Numerik (IAN)
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2019
• 资助国家: 德国
• 起止时间: 2018-12-31 至 2022-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/411046898?language=en
• 关键词: Simulation; analysis; temporal; multiscale; problems

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.

本研究计画旨在分析包含偏微分方程的时间多尺度问题。许多应用描述了长期效应，例如材料风化、由于原子缺陷（例如杂质）导致的材料断裂或生物图案形成和生长过程。这些现象往往受到重要的短尺度效应的影响，用传统的方法模拟这些过程是不可能的。动脉粥样硬化斑块的形成是一个缓慢的过程，需要数月。然而，它受到脉动血流的强烈影响，这将需要小于一秒的分辨率。在长时间内以如此精细的分辨率直接数值模拟复杂的流动问题显然是不可能的。我们将开发和分析时间上的多尺度方法，该方法基于对快速过程进行平均，从而可以考虑有效的长期问题。通常，我们可以引入一个标度参数来表示快标度和慢标度之间的关系。 我们将研究时间多尺度问题的解收敛到更简单的平均长期问题的解。收敛性将被衡量的尺度parameter.In本项目的第二部分，我们将设计和实现数值逼近方案的时间多尺度问题的有效模拟。这些数值工具的目的是近似解决平均长期问题。数值方法将基于有限元法对偏微分方程进行空间离散化，基于伽辽金法对时间离散化。为了获得有效的仿真工具，我们将基于空间时间自适应的离散化。这两个部分是共同努力进行的。为了设计数值逼近工具，我们必须知道所涉及的方程的分析性质，以便可以设计正确的平均方案。数值实验将有助于分析提供第一印象预期的收敛速度。我们开发的数值方案的一个普遍的字符，而不限于具体的应用。所有方法将在有限元软件库Gascoigne中实现，并作为开源项目发布，以便为各种相关的多尺度问题提供新的发现。时间多尺度偏微分方程问题的数学研究是一个具有挑战性的课题。到目前为止，只有很少的结果。