We address fundamental aspects in the approximation theory of vector-valued finite element methods, using finite element exterior calculus as a unifying framework. We generalize the Clément interpolant and the Scott-Zhang interpolant to finite element differential forms, and we derive a broken Bramble-Hilbert lemma. Our interpolants require only minimal smoothness assumptions and respect partial boundary conditions. This permits us to state local error estimates in terms of the mesh size. Our theoretical results apply to curl-conforming and divergence-conforming finite element methods over simplicial triangulations.
Keywords: Broken Bramble-Hilbert lemma, finite element exterior calculus, Clément interpolant, Scott-Zhang interpolant
@article{M2AN_2021__55_5_2075_0,
author = {Gawlik, Evan and Holst, Michael J. and Licht, Martin W.},
title = {Local finite element approximation of {Sobolev} differential forms},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {2075--2099},
year = {2021},
publisher = {EDP-Sciences},
volume = {55},
number = {5},
doi = {10.1051/m2an/2021034},
mrnumber = {4319601},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2021034/}
}
TY - JOUR AU - Gawlik, Evan AU - Holst, Michael J. AU - Licht, Martin W. TI - Local finite element approximation of Sobolev differential forms JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2021 SP - 2075 EP - 2099 VL - 55 IS - 5 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2021034/ DO - 10.1051/m2an/2021034 LA - en ID - M2AN_2021__55_5_2075_0 ER -
%0 Journal Article %A Gawlik, Evan %A Holst, Michael J. %A Licht, Martin W. %T Local finite element approximation of Sobolev differential forms %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2021 %P 2075-2099 %V 55 %N 5 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/m2an/2021034/ %R 10.1051/m2an/2021034 %G en %F M2AN_2021__55_5_2075_0
Gawlik, Evan; Holst, Michael J.; Licht, Martin W. Local finite element approximation of Sobolev differential forms. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 55 (2021) no. 5, pp. 2075-2099. doi: 10.1051/m2an/2021034
, and , Finite element exterior calculus, homological techniques, and applications. Acta Numerica 15 (2006) 1–155. | MR | Zbl | DOI
, and , Geometric decompositions and local bases for spaces of finite element differential forms. Comput. Methods Appl. Mech. Eng. 198 (2009) 1660–1672. | MR | Zbl | DOI
, and , Finite element exterior calculus: from Hodge theory to numerical stability. Bull. Am. Math. Soc. 47 (2010) 281–354. | MR | Zbl | DOI
and , A note on interpolation, best approximation, and the saturation property. Numerische Mathematik 131 (2015) 199–203. | MR | DOI
, and , Moment equations for the mixed formulation of the Hodge Laplacian with stochastic loading term. IMA J. Numer. Anal. 34 (2014) 1328–1360. | MR | Zbl | DOI
and , and pointwise a posteriori error estimates for FEM for elliptic PDEs on surfaces. IMA J. Numer. Anal. 35 (2015) 1199–1227. | MR | DOI
, , Equivalence of local-best and global-best approximations in . HAL Preprint: hal-02736200 (2020). | MR
, Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension. Numerische Mathematik 107 (2007) 87–106. | MR | Zbl | DOI
and , Poincaré-Friedrichs inequalities of complexes of discrete distributional differential forms. BIT Numer. Math. 60 (2020) 345–371. | MR | DOI
, and , Topics in structure-preserving discretization. Acta Numerica 20 (2011) 1. | MR | Zbl | DOI
and , Smoothed projections in finite element exterior calculus. Math. Comput. 77 (2008) 813–829. | MR | Zbl | DOI
, Approximation by finite element functions using local regularization, Revue française d’automatique, informatique, recherche opérationnelle. Anal. numérique 9 (1975) 77–84. | MR | Zbl | Numdam | DOI
and , A posteriori error estimates for finite element exterior calculus: The de Rham complex. Found. Comput. Math. 12 (2014) 1–35. | MR | Zbl
, and , Hitchhiker’s guide to the fractional Sobolev spaces. Bull. des sciences mathématiques 136 (2012) 521–573. | MR | Zbl | DOI
and , Polynomial approximation of functions in Sobolev spaces. Math. Comput. 34 (1980) 441–463. | MR | Zbl | DOI
, , and , Equivalence of local- and global-best approximations, a simple stable local commuting projector, and optimal approximation estimates in . ArXiv e-prints [] (2019). | arXiv | MR
and , Mollification in strongly Lipschitz domains with application to continuous and discrete de Rham complexes. Comput. Methods Appl. Math. 16 (2016) 51–75. | MR | DOI
and , Finite element quasi-interpolation and best approximation. ESAIM: M2AN 51 (2017) 1367–1385. | MR | Numdam | DOI
and , Local bounded cochain projections. Math. Comput. 83 (2014) 2631–2656. | MR | Zbl | DOI
and , Magnetostatic and electrostatic problems in inhomogeneous anisotropic media with irregular boundary and mixed boundary conditions. Math. Models and Methods Appl. Sci. 7 (1997) 957–991. | MR | Zbl | DOI
, An analysis of the convergence of mixed finite element methods. RAIRO. Anal. numérique 11 (1977) 341–354. | MR | Zbl | Numdam | DOI
, and , Hodge decompositions with mixed boundary conditions and applications to partial differential equations on Lipschitz manifolds. J. Math. Sci. 172 (2011) 347–400. | MR | Zbl | DOI
, and , Differential forms on Lipschitz manifolds. Siberian Math. J. 23 (1982) 151–161. | MR | Zbl | DOI
and , Partial expansion of a Lipschitz domain and some applications. Fron. Math. China 7 (2011) 1–24. | MR | Zbl
, Finite elements in computational electromagnetism. Acta Numerica 11 (2002) 237–339. | MR | Zbl | DOI
, and , Nonlinear Hodge theory on manifolds with boundary. Annali di Matematica pura ed applicata 177 (1999) 37–115. | MR | Zbl | DOI
, and , On the regularity of differential forms satisfying mixed boundary conditions in a class of Lipschitz domains. Ind. Univ. Math. J. 58 (2009) 2043–2072. | MR | Zbl | DOI
, A compactness result for vector fields with divergence and curl in involving mixed boundary conditions. Appl. Anal. 66 (1997) 189–203. | MR | Zbl | DOI
, Regularity of weak solutions of Maxwell’s equations with mixed boundary-conditions. Math. Methods Appl. Sci. 22 (1999) 1255–1274. | MR | Zbl | DOI
, Smoothed projections and mixed boundary conditions. Math. Comput. 88 (2019) 607–635. | MR | DOI
, Smoothed projections over weakly Lipschitz domains. Math. Comput. 88 (2019) 179–210. | MR | DOI
, Complexes of discrete distributional differential forms and their homology theory. Found. Comput. Math. 17 (2017) 1085–1122. | MR | DOI
, On the a priori and a posteriori error analysis in finite element exterior calculus, Ph.D. thesis. Dissertation, Department of Mathematics, University of Oslo, Norway (2017).
, and , Traces of differential forms on Lipschitz domains, the boundary de Rham complex, and Hodge decompositions. J. Funct. Anal. 190 (2002) 339–417.
, On a BPX-preconditioner for P1 elements. Comput. 51 (1993) 125–133. | MR | Zbl | DOI
, A posteriori error estimates for Maxwell equations. Math. Comput. 77 (2008) 633–649. | MR | Zbl | DOI
, theory of differential forms on manifolds. Trans. Am. Math. Soc. 347 (1995) 2075–2096. | MR | Zbl
and , Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comput. 54 (1990) 483–493. | MR | Zbl | DOI
, Generalized Sobolev spaces and their applications to boundary value problems of partial differential equations. Gos. Ped. Inst. Ucep. Zap 197 (1958) 54–112. | MR | Zbl
, Approximating gradients with continuous piecewise polynomial functions. Found. Comput. Math. 16 (2016) 723–750. | MR | DOI
Cité par Sources :
