Towards Robust Adaptive Methods for Singularly Perturbed Problems

针对奇异扰动问题的鲁棒自适应方法

• 批准号: 260814742
• 负责人: Professor Dr. Rüdiger Verfürth
• 金额: --
• 依托单位: Lehrstuhl Numerische Mathematik, insbesondere Numerik partieller Differentialgleichungen
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2014
• 资助国家: 德国
• 起止时间: 2013-12-31 至 2016-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/260814742?language=en
• 关键词: Towards; Robust; Adaptive; Methods; Singularly

Adaptive discretization methods are an indispensable tool for the efficient solution of problems in technics and natural sciences. The goal is to obtain a (nearly) optimal balance of cost to accuracy of the discrete solution. The development of of such methods is linked to a posteriori error estimation and adaptive approximation. Many practical problems are modeled by families of partial differential equations that contain parameters and which give rise to singularly perturbed problems. The balance of cost to accuracy should then be robust in the sense that it holds uniformly in the whole range of parameters. The proposed project aims at developing robust adaptive methods for such singularly perturbed problems. Starting from recent results of Tantardini, Veeser und Verfürth (submitted 2013) for singularly perturbed reaction-diffusion problems we intend to investigate in particular singularly perturbed convection-diffusion equations and problems with large jumps in the diffusivity.

自适应离散化方法是有效地解决技术和自然科学问题所不可缺少的工具。目标是在离散解的成本与精度之间取得(几乎)最优的平衡。这种方法的发展与后验误差估计和自适应逼近有关。许多实际问题都是由含有参数的偏微分方程族来模拟的，这类偏微分方程族产生了奇异摄动问题。因此，成本与精度之间的平衡应该是稳健的，因为它在整个参数范围内都是一致的。该项目旨在为这类奇异摄动问题开发稳健的自适应方法。从Tantardini，Veeser und Verfürth(2013年提交)关于奇摄动反应扩散问题的最新结果出发，我们打算特别研究奇摄动对流扩散方程和扩散系数具有大跳跃的问题。

项目成果

Robust Localization of the Best Error with Finite Elements in the Reaction-Diffusion Norm

反应扩散范数中有限元最佳误差的鲁棒定位

• DOI: 10.1007/s00365-015-9291-5
• 发表时间: 2015
• 期刊: Constructive Approximation
• 影响因子: 2.7
• 作者: F. Tantardini;A. Veeser;R. Verfürth
• 通讯作者: R. Verfürth