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The stability in \(L_ p\) and \(W^ 1_ p\) of the \(L_ 2\)-projection onto finite element function spaces. (English) Zbl 0637.41034
The stability in \(L_ p\) and \(W^ 1_ p\), for \(1\leq p\leq \infty\), of the \(L_ 2\)-projection onto some standard finite element subspaces \(V_ h\) is shown. In the one-dimensional case the authors prove \(L_ p\) stability without any restriction on partition and \(W^ 1_ p\) stability, (p,1), under a quite weak assumption on the partition, depending on p. It is also shown that some restriction on the partition is needed for stability if \(p>1.\)
In the case of two-dimensional polygonal domain and \(L_ p\) and \(W^ 1_ p\) stability is shown for the \(L_ 2\)-projection onto standard piecewise polynomial spaces of Lagrangian type. Less than quasi- uniformity of the triangulations that define the subspaces \(V_ h\) is required.
Reviewer: C.Simerská

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
41A15 Spline approximation
41A50 Best approximation, Chebyshev systems
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
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