On the stability of the L2 projection in H1(Ω)

Authors:
James H. Bramble;Joseph E. Pasciak;Olaf Steinbach
Affiliations:
Department of Mathematics, Texas A & M University, College Station, Texas;Department of Mathematics, Texas A & M University, College Station, Texas;Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Venue:
Mathematics of Computation
Year:
2002

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Abstract

We prove the stability in H1(Ω) of the L2 projection onto a family of finite element spaces of conforming piecewise linear functions satisfying certain local mesh conditions. We give explicit formulae to check these conditions for a given finite element mesh in any number of spatial dimensions. In particular, stability of the L2 projection in H1(Ω) holds for locally quasiuniform geometrically refined meshes as long as the volume of neighboring elements does not change too drastically.