Towards Robust Adaptive Methods for Singularly Perturbed Problems

自适应离散化方法是有效解决技术和自然科学问题不可缺少的工具。目标是获得离散解的代价与精度的（近乎）最优平衡。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包含参数的偏微分方程族来建模的，这些偏微分方程族会引起奇摄动问题。成本与精度的平衡应该是鲁棒的，因为它在整个参数范围内保持一致。提出的项目旨在为此类奇摄动问题开发鲁棒自适应方法。从Tantardini， Veeser和verf<e:1> rth（2013年提交）关于奇摄动反应扩散问题的最新结果开始，我们打算特别研究奇摄动对流扩散方程和扩散率大跳跃的问题。

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.