Finite element quasi-interpolation and best approximation
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 51 (2017) no. 4, pp. 1367-1385.

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
Mots clés : 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.
@article{M2AN_2017__51_4_1367_0,
     author = {Ern, Alexandre and Guermond, Jean-Luc},
     title = {Finite element quasi-interpolation and best approximation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1367--1385},
     publisher = {EDP-Sciences},
     volume = {51},
     number = {4},
     year = {2017},
     doi = {10.1051/m2an/2016066},
     mrnumber = {3702417},
     zbl = {1378.65041},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2016066/}
}
TY  - JOUR
AU  - Ern, Alexandre
AU  - Guermond, Jean-Luc
TI  - Finite element quasi-interpolation and best approximation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2017
SP  - 1367
EP  - 1385
VL  - 51
IS  - 4
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2016066/
DO  - 10.1051/m2an/2016066
LA  - en
ID  - M2AN_2017__51_4_1367_0
ER  - 
%0 Journal Article
%A Ern, Alexandre
%A Guermond, Jean-Luc
%T Finite element quasi-interpolation and best approximation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2017
%P 1367-1385
%V 51
%N 4
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2016066/
%R 10.1051/m2an/2016066
%G en
%F M2AN_2017__51_4_1367_0
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. http://archive.numdam.org/articles/10.1051/m2an/2016066/

Y. Achdou, C. Bernardi and F. Coquel, A priori and a posteriori analysis of finite volume discretizations of Darcy’s equations. Numer. Math. 96 (2003) 17–42. | DOI | MR | Zbl

D.N. Arnold, R.S. Falk and R. Winther, Finite element exterior calculus, homological techniques, and applications. Acta Numer. 15 (2006) 1–155. | DOI | MR | Zbl

C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35 (1998) 1893–1916. | DOI | MR | Zbl

J.H. Bramble and S.R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation. SIAM J. Numer. Anal. 7 (1970) 112–124. | DOI | MR | Zbl

E. Burman and A. Ern, Continuous interior penalty hp-finite element methods for advection and advection-diffusion equations. Math. Comput. 76 (2007) 1119–1140. | DOI | MR | Zbl

M. Campos Pinto and E. Sonnendrücker, Gauss-compatible Galerkin schemes for time-dependent Maxwell equations. Math. Comput. 302 (2016) 2651–2685. | DOI | MR | Zbl

P. Ciarlet Jr,, Analysis of the Scott-Zhang interpolation in the fractional order Sobolev spaces. J. Numer. Math. 21 (2013) 173–180. | MR | Zbl

P.G. Ciarlet, The finite element method for elliptic problems, Vol. 40 of Classics in Applied Mathematics. Society for Industrial and Applied Mathematics (SIAM). Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. Philadelphia, PA (2002). | MR | Zbl

P. Clément, Approximation by finite element functions using local regularization. RAIRO: Anal. Numer. 9 (1975) 77–84. | Numdam | MR | Zbl

B. Cockburn, G. Kanschat and D. Schötzau, A note on discontinuous Galerkin divergence-free solutions of the Navier-Stokes equations. J. Sci. Comput. 31 (2007) 61–73. | DOI | MR | Zbl

D. Di Pietro and A. Ern, Mathematical aspects of discontinuous Galerkin methods. Vol. 69 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer, Heidelberg (2012). | MR | Zbl

T. Dupont and R. Scott, Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34 (1980) 441–463. | DOI | MR | Zbl

A. Ern and J.-L. Guermond, Theory and practice of finite elements. Vol. 159 of Appl. Math. Sci. Springer-Verlag, New York (2004). | MR | Zbl

A. Ern, A.F. Stephansen and M. Vohralík, Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems. J. Comput. Appl. Math. 234 (2010) 114–130. | DOI | MR | Zbl

V. Girault and J.-L. Lions, Two-grid finite-element schemes for the transient Navier-Stokes problem. ESAIM: M2AN 35 (2001) 945–980. | DOI | Numdam | MR | Zbl

P. Grisvard, Elliptic problems in nonsmooth domains. Vol. 24 of Monographs and Studies in Mathematics. Pitman (Advanced Publishing Program), Boston, MA (1985). | MR | Zbl

N. Heuer, On the equivalence of fractional-order Sobolev semi-norms. J. Math. Anal. Appl. 417 (2014) 505–518. | DOI | MR | Zbl

O.A. Karakashian and F. Pascal, A posteriori error estimates for a discontinuous Galerkin approximation of second-order elliptic problems. SIAM J. Numer. Anal. 41 (2003) 2374–2399. | DOI | MR | Zbl

C.B. Morrey, Jr, Mettre en romain : Multiple integrals in the calculus of variations. Die Grundlehren der Mathematischen Wissenschaften, Band 130. Springer-Verlag New York, Inc., New York (1966). | MR | Zbl

P. Oswald, On a BPX-preconditioner for P1 elements. Computing 51 (1993) 125–133. | DOI | MR | Zbl

A.C. Ponce and J. Van Schaftingen, The continuity of functions with N-th derivative measure. Houston J. Math. 33 (2007) 927–939. | MR | Zbl

J. Schöberl and C. Lehrenfeld, Domain decomposition preconditioning for high order hybrid discontinuous Galerkin methods on tetrahedral meshes. In Advanced finite element methods and applications. Vol. 66 of Lect. Notes Appl. Comput. Mech. Springer, Heidelberg (2013) 27–56. | MR | Zbl

R.L. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. | DOI | MR | Zbl

L. Tartar, An introduction to Sobolev spaces and interpolation spaces. Vol. 3 of Lecture Notes of the Unione Matematica Italiana. Springer, Berlin; UMI, Bologna (2007). | MR | Zbl

Cité par Sources :