各向异性网格下Oseen问题的稳定化有限元方法

求解Oseen问题的混合有限元方法通常有两个障碍，其一，混合有限元空间要满足LBB条件；其二，若问题具有对流占优性质，采用标准的Galerkin离散方法会导致近似解的非物理震荡。稳定化方法是扫除这两个障碍的有效工具。到目前为止，大多数稳定化方法都采用各向同性网格，但对于边界层、激波、交界面及边奇异等问题的数值算法，采用各向异性网格不仅可以更好的逼近真解的物理性态，还能够大大减少自由度以节省计算量。本项目将建立各向异性网格下求解Oseen问题的稳定化有限元方法；研究相应的自适应算法，包括h方法、r方法以及hr方法；进而编制程序来解决实际物理问题。

There are two main obstacles when solving Oseen problems. First, mixed finite element spaces need to satisfy LBB condition. Second, standard Galerkin discretization will cause non-physical oscillations for problem with convection dominated property. Stabilized method is an efficient way to solve the two obstacles above. So far, most stabilized methods are based on isotropic meshes. For those problems having boundary layers, shock waves, interface, and edge singularities, e.t., numerical methods using anisotropic meshes will not only approximate physics properties of the true solution better, but also reduce the freedom of unknowns and thus save computational load. This project will establish stabilized method for Oseen problems under anisotropic mesh, study corresponding adaptive strategy which includes h method, r method and hr method, and then program to solve pratical physical problems.