Analysis and stabilisation methods for singularly perturbed problems and systems

当奇摄动问题在均匀网格上用标准方法数值求解时，数值解经常出现虚假振荡。另一方面，如果使用稳定化方法和层适应网格，精确解可以很好地逼近。对于相对简单的问题，如单位正方形中的对流扩散，解的行为和层适应网格的构造是很好理解的。对于更复杂的问题，如Oseen方程，这是一个悬而未决的问题。因此，这个建议的一个主要目标是调查问题的解决方案的行为，如Oseen方程与合适的稳定方法。这是一个重要的里程碑，以更好地了解Navier-Stokes方程的解决方案的行为。

When singularly perturbed problems are solved numerically on a uniform mesh with standard methods, the numerical solution often exhibits spurious oscillations. On the other hand, if stabilised methods and layer-adapted meshes are used, the exact solutions are well approximated.For relatively simple problems like convection diffusion in the unit square the behaviour of solutions and the construction of layer adapted meshes is well understood. For more complex problems like the Oseen equations, this is an open question. Thus, a main goal of this proposal is to investigate the solution behaviour of problems like the Oseen-equations together with suitable stabilisation methods. This is an important milestone to a better understanding of the solution behaviour of Navier-Stokes equations.