Geodesic
Finite element quasi-interpolation and best approximation
Ern, Alexandre1  ; Guermond, Jean-Luc2 
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.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1367-1385

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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 L p -norm assuming regularity in the fractional Sobolev spaces W r,p , where p∈[1,∞] and the smoothness index r can be arbitrarily close to zero. The operator is stable in L 1 , leaves the corresponding finite element space point-wise invariant, and can be modified to handle homogeneous boundary conditions. The theory is illustrated on H 1 -, 𝐇(curl)- and 𝐇(div)-conforming spaces.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2016066
Classification : 65D05, 65N30, 41A65
Keywords: Quasi-interpolation, finite elements, best approximation

Ern, Alexandre 1 ; Guermond, Jean-Luc 2

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. 
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     title = {Finite element quasi-interpolation and best approximation},
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     publisher = {EDP-Sciences},
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Ern, Alexandre; Guermond, Jean-Luc. Finite element quasi-interpolation and best approximation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1367-1385. doi: 10.1051/m2an/2016066

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