Adaptive Uzawa algorithm for the Stokes equation
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 6, pp. 1841-1870.

Based on the Uzawa algorithm, we consider an adaptive finite element method for the Stokes system. We prove linear convergence with optimal algebraic rates for the residual estimator (which is equivalent to the total error), if the arising linear systems are solved iteratively, e.g., by PCG. Our analysis avoids the use of discrete efficiency of the estimator. Unlike prior work, our adaptive Uzawa algorithm can thus avoid to discretize the given data and does not rely on an interior node property for the refinement.

Reçu le :
Accepté le :
DOI : 10.1051/m2an/2019039
Classification : 65N30, 65N50, 65N15, 41A25
Mots-clés : Adaptive finite element method, optimal convergence, Uzawa algorithm, Stokes equation
Di Fratta, Giovanni 1 ; Führer, Thomas 1 ; Gantner, Gregor 1 ; Praetorius, Dirk 1

1
@article{M2AN_2019__53_6_1841_0,
     author = {Di Fratta, Giovanni and F\"uhrer, Thomas and Gantner, Gregor and Praetorius, Dirk},
     title = {Adaptive {Uzawa} algorithm for the {Stokes} equation},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1841--1870},
     publisher = {EDP-Sciences},
     volume = {53},
     number = {6},
     year = {2019},
     doi = {10.1051/m2an/2019039},
     mrnumber = {4019762},
     zbl = {1434.65248},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2019039/}
}
TY  - JOUR
AU  - Di Fratta, Giovanni
AU  - Führer, Thomas
AU  - Gantner, Gregor
AU  - Praetorius, Dirk
TI  - Adaptive Uzawa algorithm for the Stokes equation
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2019
SP  - 1841
EP  - 1870
VL  - 53
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2019039/
DO  - 10.1051/m2an/2019039
LA  - en
ID  - M2AN_2019__53_6_1841_0
ER  - 
%0 Journal Article
%A Di Fratta, Giovanni
%A Führer, Thomas
%A Gantner, Gregor
%A Praetorius, Dirk
%T Adaptive Uzawa algorithm for the Stokes equation
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2019
%P 1841-1870
%V 53
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2019039/
%R 10.1051/m2an/2019039
%G en
%F M2AN_2019__53_6_1841_0
Di Fratta, Giovanni; Führer, Thomas; Gantner, Gregor; Praetorius, Dirk. Adaptive Uzawa algorithm for the Stokes equation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 53 (2019) no. 6, pp. 1841-1870. doi : 10.1051/m2an/2019039. http://www.numdam.org/articles/10.1051/m2an/2019039/

E. Bänsch, P. Morin and R.H. Nochetto, An adaptive Uzawa FEM for the Stokes problem: convergence without the inf-sup condition. SIAM J. Numer. Anal. 40 (2002) 1207–1229. | DOI | MR | Zbl

R. Becker and S. Mao, Quasi-optimality of adaptive nonconforming finite element methods for the Stokes equations. SIAM J. Numer. Anal. 49 (2011) 970–991. | DOI | MR | Zbl

A. Bespalov, A. Haberl and D. Praetorius, Adaptive FEM with coarse initial mesh guarantees optimal convergence rates for compactly perturbed elliptic problems. Comput. Methods Appl. Mech. Eng. 317 (2017) 318–340. | DOI | MR | Zbl

P. Binev, Tree approximation for hp-adaptivity. SIAM J. Numer. Anal. 56 (2018) 3346–3357. | DOI | MR | Zbl

P. Binev and R. Devore, Fast computation in adaptive tree approximation. Numer. Math. 97 (2004) 193–217. | DOI | MR | Zbl

P. Binev, W. Dahmen and R. Devore, Adaptive finite element methods with convergence rates. Numer. Math. 97 (2004) 219–268. | DOI | MR | Zbl

P. Binev, W. Dahmen, R. Devore and P. Petrushev, Approximation classes for adaptive methods. Serdica Math. J. 28 (2002) 391–416. | MR | Zbl

J.H. Bramble, A proof of the inf–sup condition for the Stokes equations on Lipschitz domains. Math. Models Methods Appl. Sci. 13 (2003) 361–371. | DOI | MR | Zbl

C. Carstensen, D. Peterseim and H. Rabus, Optimal adaptive nonconforming FEM for the Stokes problem. Numer. Math. 123 (2013) 291–308. | DOI | MR | Zbl

C. Carstensen, M. Feischl, M. Page and D. Praetorius, Axioms of adaptivity. Comput. Math. Appl. 67 (2014) 1195–1253. | DOI | MR | Zbl

J.M. Cascon, C. Kreuzer, R.H. Nochetto and K.G. Siebert, Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46 (2008) 2524–2550. | DOI | MR | Zbl

S. Dahlke, W. Dahmen and K. Urban, Adaptive wavelet methods for saddle point problems—optimal convergence rates. SIAM J. Numer. Anal. 40 (2002) 1230–1262. | DOI | MR | Zbl

C. Erath, G. Gantner and D. Praetorius, Optimal convergence behavior of adaptive FEM driven by simple (h − h/2)-type error estimators (2018) Preprint arXiv: 1805.00715. | MR | Zbl

M. Feischl, Optimal adaptivity for a standard finite element method for the Stokes problem (2017) Preprint arXiv: 1710.08289. | MR

T. Führer, A. Haberl, D. Praetorius and S. Schimanko, Adaptive BEM with inexact PCG solver yields almost optimal computational costs. Numer. Math. (2018) published online first. | MR | Zbl

D. Gallistl, M. Schedensack and R. Stevenson, A remark on newest vertex bisection in any space dimension. Comput. Methods Appl. Math. 14 (2014) 317–320. | DOI | MR | Zbl

T. Gantumur, On the convergence theory of adaptive mixed finite element methods for the Stokes problem (2014). Preprint arXiv: 1403.0895

T. Gantumur, Convergence rates of adaptive methods, Besov spaces, and multilevel approximation. Found. Comput. Math. 17 (2017) 917–956. | DOI | MR | Zbl

F.D. Gaspoz and P. Morin, Convergence rates for adaptive finite elements. IMA J. Numer. Anal. 29 (2008) 917–936. | DOI | MR | Zbl

G.H. Golub and C.F. Van Loan, Matrix Computations. 4th ed., Johns Hopkins University Press, Baltimore, MD, 2013. | MR | Zbl

J. Hu and J. Xu, Convergence and optimality of the adaptive nonconforming linear element method for the Stokes problem. J. Sci. Comput. 55 (2013) 125–148. | DOI | MR | Zbl

M. Karkulik, D. Pavlicek and D. Praetorius, On 2D newest vertex bisection: optimality of mesh-closure and H1-stability of L2-projection. Constr. Approx. 38 (2013) 213–234. | DOI | MR | Zbl

Y. Kondratyuk, Adaptive Finite Element Algorithms for the Stokes Problem: COnvergence Rates and Optimal Computational Complexity, Department of Mathematics, Utrecht University, The Netherlands, Preprint, 2006, p. 1346.

Y. Kondratyuk and R. Stevenson, An optimal adaptive finite element method for the Stokes problem. SIAM J. Numer. Anal. 46 (2008) 747–775. | DOI | MR | Zbl

P. Morin, K.G. Siebert and A. Veeser, A basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18 (2008) 707–737. | DOI | MR | Zbl

K.G. Siebert, A convergence proof for adaptive finite elements without lower bound. IMA J. Numer. Anal. 31 (2010) 947–970. | DOI | MR | Zbl

R. Stevenson, Optimality of a standard adaptive finite element method. Found. Comput. Math. 7 (2007) 245–269. | DOI | MR | Zbl

R. Stevenson, The completion of locally refined simplicial partitions created by bisection. Math. Comput. 77 (2008) 227–241. | DOI | MR | Zbl

Cité par Sources :