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的不可缺少的工具。我们的目标是获得一个（接近）最佳的平衡成本的离散解决方案的准确性。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模拟的，这些偏微分方程族会引起奇摄动问题。成本与准确性的平衡应该是稳健的，因为它在整个参数范围内保持一致。拟议的项目旨在为这种奇异摄动问题开发鲁棒自适应方法。从最近的结果Tantardini，Veeser und Verfürth（提交2013年）奇摄动反应扩散问题，我们打算特别研究奇摄动对流扩散方程和问题的大跳跃的扩散。

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.