Abstract
This paper establishes a foundation of non-conforming boundary elements. We present a discrete weak formulation of hypersingular integral operator equations that uses Crouzeix–Raviart elements for the approximation. The cases of closed and open polyhedral surfaces are dealt with. We prove that, for shape regular elements, this non-conforming boundary element method converges and that the usual convergence rates of conforming elements are achieved. Key ingredient of the analysis is a discrete Poincaré–Friedrichs inequality in fractional order Sobolev spaces. A numerical experiment confirms the predicted convergence of Crouzeix–Raviart boundary elements.
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References
Bespalov A., Heuer N.: The hp version of the boundary element method with quasi-uniform meshes in three dimensions. ESAIM Math. Model. Numer. Anal. 42, 821–849 (2008)
Brenner S.: Poincaré–Friedrichs inequalities for piecewise H 1 functions. SIAM J. Numer. Anal. 41, 306–324 (2003)
Buffa A., Ciarlet P. Jr.: On traces for functional spaces related to Maxwell’s equations I. An integration by parts formula in Lipschitz polyhedra. Math. Methods Appl. Sci. 24, 9–30 (2001)
Buffa A., Costabel M., Sheen D.: On traces for H(curl, Ω) in Lipschitz domains. J. Math. Anal. Appl. 276, 845–867 (2002)
Crouzeix M., Raviart P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I, Rev. Française Automat. Inf. Recherche Opérationnelle Sér. Rouge 7, 33–75 (1973)
Ervin V.J., Heuer N., Stephan E.P.: On the hp version of the boundary element method for Symm’s integral equation on polygons. Comput. Methods Appl. Mech. Eng. 110, 25–38 (1993)
Gatica, G., Healey, M., Heuer, N.: The boundary element method with Lagrangian multipliers. Numer. Methods Partial Diff. Eq. (2008, in press). doi:10.1002/num.20401
Heuer N.: Additive Schwarz method for the p version of the boundary element method for the single layer potential operator on a plane screen. Numer. Math. 88, 485–511 (2001)
Hsiao G.C., Wendland W.L.: A finite element method for some integral equations of the first kind. J. Math. Anal. Appl. 58, 449–481 (1977)
Lions J.-L., Magenes E.: Non-Homogeneous Boundary Value Problems and Applications I. Springer, New York (1972)
McLean W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, London (2000)
Nédélec J.-C., Planchard J.: Une méthode variationnelle d’éléments finis pour la résolution numérique d’un problème extérieur dans R 3. RAIRO 7(R-3), 105–129 (1973)
Stephan E.P.: A boundary integral equation method for three-dimensional crack problems in elasticity. Math. Methods Appl. Sci. 8, 609–623 (1986)
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Norbert Heuer is supported by Fondecyt-Chile under grant no. 1080044. F.-J. Sayas is partially supported by MEC-FEDER Project MTM2007-63204 and Gobierno de Aragón (Grupo Consolidado PDIE).