Analysis and stabilisation methods for singularly perturbed problems and systems

用标准方法在均匀网格上数值求解奇异摄动问题时，数值解往往会出现伪振荡。另一方面，如果使用稳定化方法和层适应网格，精确解是非常接近的。对于相对简单的问题，如单位正方形内的对流扩散，解的性质和层适应网格的构造是很好地理解的。对于像奥西方程这样更复杂的问题，这是一个悬而未决的问题。因此，这一建议的一个主要目标是研究像Osee方程这样的问题的解的行为，以及适当的稳定化方法。这对于更好地理解Navier-Stokes方程的解行为是一个重要的里程碑。

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.