Which Adaptive FE Spaces Allow for a H1-Stable L2-Projection?
Carsten Carstensen (Uni Kiel)
Within automatic mesh-refining algorithms, the focus is on
good properties of the designed mesh such as regularity (no hanging
shape regularity (no degenerated angles or aspect ratios).
This presentation is, in addition, concerned with a stability
property of the generated finite element spaces.
The presentation briefly reports on a recent result on
the uniform H1-stability of the L2-projection onto the discrete
The affirmative result states that any red-green-blue-refinement
conforming piecewise affine spaces on triangles
where the bound for the L2-projection
only depends on the coarse mesh. The underlying criterion
on the commutator property of local mass matrices with certain weighted
diagonal matrices is based on recent work of Bramble, Pasciak, and
The diagonal matrices measure the refinement levels and the criterion
limits the refinement which should not be too high - as expected from
due to Crouzeix and Thomee.
Unfortunatly, the proof fails for parallelograms or three-dimensional
situations. The presentation addresses those general situations and
discusses extra conditions on the refinement procedure in order to keep
the L2-projection onto the finite element spaces uniformly