Geodesic
Adaptive Crouzeix–Raviart boundary element method
Heuer, Norbert1 ; Karkulik, Michael1
1 Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna, 4860 Santiago, Chile.
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 1193-1217.

Voir la notice de l'article provenant de la source Numdam

For the nonconforming Crouzeix–Raviart boundary elements from [N. Heuer and F.-J. Sayas, Numer. Math. 112 (2009) 381–401], we develop and analyze a posteriori error estimators based on the h-h/2 methodology. We discuss the optimal rate of convergence for uniform mesh refinement, and present a numerical experiment with singular data where our adaptive algorithm recovers the optimal rate while uniform mesh refinement is sub-optimal. We also discuss the case of reduced regularity by standard geometric singularities to conjecture that, in this situation, non-uniformly refined meshes are not superior to quasi-uniform meshes for Crouzeix–Raviart boundary elements.

Reçu le :
DOI : 10.1051/m2an/2015003
Classification : 65N30, 65N38, 65N50, 65R20
Keywords: Boundary element method, adaptive algorithm, nonconforming method, a posteriori error estimation

Heuer, Norbert 1 ; Karkulik, Michael 1

1 Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna, 4860 Santiago, Chile. 
@article{M2AN_2015__49_4_1193_0,
     author = {Heuer, Norbert and Karkulik, Michael},
     title = {Adaptive {Crouzeix{\textendash}Raviart} boundary element method},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1193--1217},
     publisher = {EDP-Sciences},
     volume = {49},
     number = {4},
     year = {2015},
     doi = {10.1051/m2an/2015003},
     mrnumber = {3371908},
     zbl = {1326.65166},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015003/}
}
TY  - JOUR
AU  - Heuer, Norbert
AU  - Karkulik, Michael
TI  - Adaptive Crouzeix–Raviart boundary element method
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2015
SP  - 1193
EP  - 1217
VL  - 49
IS  - 4
PB  - EDP-Sciences
UR  - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015003/
DO  - 10.1051/m2an/2015003
LA  - en
ID  - M2AN_2015__49_4_1193_0
ER  - 
%0 Journal Article
%A Heuer, Norbert
%A Karkulik, Michael
%T Adaptive Crouzeix–Raviart boundary element method
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2015
%P 1193-1217
%V 49
%N 4
%I EDP-Sciences
%U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015003/
%R 10.1051/m2an/2015003
%G en
%F M2AN_2015__49_4_1193_0
Heuer, Norbert; Karkulik, Michael. Adaptive Crouzeix–Raviart boundary element method. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 49 (2015) no. 4, pp. 1193-1217. doi : 10.1051/m2an/2015003. http://geodesic.mathdoc.fr/articles/10.1051/m2an/2015003/

M. Ainsworth and J. Tinsley Oden, A posteriori error estimation in finite element analysis. Pure Appl. Math. Wiley-Interscience [John Wiley & Sons], New York (2000). | Zbl | MR

M. Aurada, M. Feischl, T. Führer, M. Karkulik and D. Praetorius, Efficiency and Optimality of Some Weighted–Residual Error Estimator for Adaptive 2D Boundary Element Methods. Comput. Methods Appl. Math. 13 (2013) 305–332. | Zbl | MR | DOI

M. Aurada, M. Feischl, T. Führer, M. Karkulik and D. Praetorius, Energy norm based error estimators for adaptive BEM for hypersingular integral equations. Appl. Numer. Math. (2014). | MR

R.E. Bank, Hierarchical bases and the finite element method. In vol. 5 of Acta Numer. Cambridge Univ. Press, Cambridge (1996) 1–43. | Zbl | MR

A. Berger, R. Scott and G. Strang, Approximate boundary conditions in the finite element method. In vol. X, Symposia Mathematica (Convegno di Analisi Numerica, INDAM, Rome, 1972). Academic Press, London (1972) 295–313. | Zbl | MR

A. Bespalov and N. Heuer, The hp-version of the boundary element method with quasi-uniform meshes in three dimensions. ESAIM: M2AN 42 (2008) 821–849. | Zbl | mathdoc-id | MR | DOI

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

A. Bonito and R.H. Nochetto, Quasi-optimal convergence rate of an adaptive discontinuous Galerkin method. SIAM J. Numer. Anal. 48 (2010) 734–771. | Zbl | MR | DOI

A. Buffa, M. Costabel and D. Sheen, On traces for 𝐇(𝐜𝐮𝐫𝐥,Ω) in Lipschitz domains. J. Math. Anal. Appl. 276 (2002) 845–867. | Zbl | MR | DOI

C. Carstensen and D. Praetorius, Averaging techniques for the a posteriori BEM error control for a hypersingular integral equation in two dimensions. SIAM J. Sci. Comput. 29 (2007) 782–810. | MR | Zbl | DOI

C. Carstensen, M. Maischak, D. Praetorius and E.P. Stephan, Residual-based a posteriori error estimate for hypersingular equation on surfaces. Numer. Math. 97 (2004) 397–425. | MR | Zbl | DOI

Ph. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 9 (1975) 77–84. | MR | Zbl | mathdoc-id

C. Domínguez and N. Heuer, A posteriori error analysis for a boundary element method with non-conforming domain decomposition. Numer. Methods Partial Differ. Eq. 30 (2014) 947–963. | MR | Zbl | DOI

W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. | MR | Zbl | DOI

W. Dörfler and R.H. Nochetto, Small data oscillation implies the saturation assumption. Numer. Math. 91 (2002) 1–12. | MR | Zbl | DOI

Ch. Erath, S. Ferraz-Leite, S. Funken and D. Praetorius, Energy norm based a posteriori error estimation for boundary element methods in two dimensions. Appl. Numer. Math. 59 (2009) 2713–2734. | MR | Zbl | DOI

V.J. Ervin and N. Heuer, An adaptive boundary element method for the exterior Stokes problem in three dimensions. IMA J. Numer. Anal. 26 (2006) 297–325. | MR | Zbl | DOI

S. Ferraz-Leite and D. Praetorius, Simple a posteriori error estimators for the h-version of the boundary element method. Computing 83 (2008) 135–162. | MR | Zbl | DOI

S. Ferraz-Leite, C. Ortner and D. Praetorius, Convergence of simple adaptive Galerkin schemes based on h-h/2 error estimators. Numer. Math. 116 (2010) 291–316. | MR | Zbl | DOI

G.N. Gatica, M. Healey and N. Heuer, The boundary element method with Lagrangian multipliers. Numer. Methods Partial Differ. Eq. 25 (2009) 1303–1319. | MR | Zbl | DOI

I.G. Graham, W. Hackbusch and S.A. Sauter, Finite elements on degenerate meshes: inverse-type inequalities and applications. IMA J. Numer. Anal. 25 (2005) 379–407. | MR | Zbl | DOI

E. Hairer, S.P. Nørsett and G. Wanner, Solving ordinary differential equations. I, Nonstiff problems. In vol. 8 of Springer Ser. Comput. Math. Springer-Verlag, Berlin (1987). | MR | Zbl

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

N. Heuer and F.-J. Sayas, Crouzeix–Raviart boundary elements. Numer. Math. 112 (2009) 381–401. | MR | Zbl | DOI

M. Karkulik, D. Pavlicek and D. Praetorius, On 2D Newest Vertex Bisection: Optimality of Mesh–Closure and H 1 -Stability of L 2 -Projection. Constr. Approx. 38 (2013) 213–234. | MR | Zbl | DOI

W. McLean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000). | MR | Zbl

J.-C. Nédélec, Integral equations with nonintegrable kernels. Int. Eq. Oper. Theory 5 (1982) 562–572. | MR | Zbl | DOI

E.P. Stephan, Boundary integral equations for screen problems in R 3 . Int. Eq. Oper. Theory 10 (1987) 257–263. | MR | Zbl

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

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

H. Triebel, Interpolation theory, function spaces, differential operators, 2nd edn. Edited by Johann Ambrosius Barth, Heidelberg (1995). | MR | Zbl

R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. B.G. Teubner, Stuttgart (1996). | Zbl

Cité par Sources :