Volume 51, Number 4, July-August 2017
|Page(s)||1367 - 1385|
|Published online||21 July 2017|
Finite element quasi-interpolation and best approximation∗
1 Université Paris-Est, CERMICS (ENPC), 77455 Marne-la-Vallée cedex 2, France.
2 Department of Mathematics, Texas A&M University 3368 TAMU, College Station, TX 77843, USA.
Received: 24 February 2016
Revised: 12 September 2016
Accepted: 12 October 2016
This paper introduces a quasi-interpolation operator for scalar- and vector-valued finite element spaces constructed on affine, shape-regular meshes with some continuity across mesh interfaces. This operator gives optimal estimates of the best approximation error in any Lp-norm assuming regularity in the fractional Sobolev spaces Wr,p, where p ∈ [ 1,∞ ] and the smoothness index r can be arbitrarily close to zero. The operator is stable in L1, leaves the corresponding finite element space point-wise invariant, and can be modified to handle homogeneous boundary conditions. The theory is illustrated on H1-, H(curl)- and H(div)-conforming spaces.
Mathematics Subject Classification: 65D05 / 65N30 / 41A65
Key words: Quasi-interpolation / finite elements / best approximation
This material is based upon work supported in part by the National Science Foundation grant DMS-1217262, by the Air Force Office of Scientific Research, USAF, under grant/contract number FA9550-15-1-0257 and by the Army Research Office under grant number W911NF-15-1-0517, Draft version, July 19, 2017
© EDP Sciences, SMAI 2017
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