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.

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
Mots-clés : 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. 
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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://www.numdam.org/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). | MR | Zbl

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. | DOI | MR | Zbl

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. | MR | Zbl

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. | MR | Zbl

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. | DOI | Numdam | 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

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

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

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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

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. | DOI | MR | Zbl

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

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

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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. | DOI | MR | Zbl

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

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

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