关于$L^2$ -投影的Sobolev稳定性和$L^p$ -稳定性
On the Sobolev and $L^p$-Stability of the $L^2$-projection
摘要：
我们证明了稳定的$L^2$ -投影到拉格朗日有限元空间的（加权）$L^p$和$W^{1,p}$ -规范的任何多项式的程度和任何空间维度在适当的条件下的网格分级。这包括$W^{1,2}$ -在二维空间中对任意多项式次数的稳定性和由最新顶点平分生成的网格。根据现实的，但在三维网格分级的假设，我们证明了$W^{1,2}$ -稳定的所有多项式次数。我们还提出了一个修改的二分法，导致更好的$W^{1,p}$稳定性。此外，我们研究了$L^2$ -投影到Crouzeix-Raviart元上的稳定性。
英文摘要：
We show stability of the $L^2$-projection onto Lagrange finite element spaces with respect to (weighted) $L^p$ and $W^{1,p}$-norms for any polynomial degree and for any space dimension under suitable conditions on the mesh grading. This includes $W^{1,2}$-stability in two space dimensions for any polynomial degree and meshes generated by newest vertex bisection. Under realistic but conjectured assumptions on the mesh grading in three dimensions we show $W^{1,2}$-stability for all polynomial degrees. We also propose a modified bisection strategy that leads to better $W^{1,p}$-stability. Moreover, we investigate the stability of the $L^2$-projection onto Crouzeix-Raviart elements.
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