On the Sobolev and L^p-Stability of the L^2-projection

08/04/2020

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by   Lars Diening, et al.

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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 assumptions on the mesh grading in three dimensions we show W^1,2-stability for all polynomial degrees greater than one. 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.