Navier-Stokes方程的局部投影稳定化方法

将Matthies，Skrzypacz和Tubiska的思想从线性的Oseen方程拓展到了非线性的Navier-Stokes方程，针对不可压缩的定常Navier-Stokes方程，提出了一种局部投影稳定化有限元方法．该方法既克服了对流占优，又绕开了inf-sup条件的限制．给出的局部投影空间既可以定义在两种不同网格上，又可以定义在相同网格上．与其他两级方法相比，定义在同一网格空间上的局部投影稳定化格式更紧凑．在同一网格上，除了给出需要bubble函数来增强的逼近空间外，还特别考虑了两种不需要用bubble函数来增强的新的空间．基于一种特殊的插值技巧，给出了稳定性分析和误差估计．最后，还列举了两个数值算例，进一步验证了理论结果的正确性．

Local Projection Stabilized Finite Element Method for the Navier-Stokes Equations

The results of Matthies, Skrzypacz and Tubiska for the Oseen problem to the Navier-Stokes problem were extended. For the stationary incompressible Navier-Stokes equations, a local projection stabilized finite element scheme was proposed. The schem eovercomes convection dominated and ameliorates the restrictiveinf-supcondition. Local projection schemes were derived not only as a two-level approach but also for pairs of spaces which were defined on the samemesh. This class of stabilized schemes uses approxmiation and projection spaces defined on the same mesh and leads to much more compact stencils than in the two-level approach. On the same mesh, bes ides the class of local projection stabilization by enrichment of the approximation spaces, two new classes of local projection stabilization of the approximation spaces which dont. need to be enriched by bubble functions are derived. Based on a special in terpolation, the stability and an optimal priorierror estimates were shown. Finally, the numerical tests and the numerical computations show that the numerical results agree with some ben chmark solutions, which further poved the correctness of the theoretical analysis.