Abstract
In this note we study a posteriori error estimates for a model problem in the symmetric coupling of boundary element and finite elements methods. Emphasis is on the use of the Poincaré-Steklov operator and its discretization which are analyzed in general for both a priori and a posteriori error estimates. Combining arguments from [6] and [9, 10] we refine the a posteriori error estimate obtained in [9, 10]. For quasi-uniform meshes on the boundary, we prove some inequality of a reverse type using techniques from [5] and [36]. This indicates efficiency of the new estimate as illustrated in a numerical example.
Zusammenfassung
In dieser Arbeit werden a posteriori Fehlerabschätzungen für ein Modellproblem der symmetrischen Kopplung von Finiten Elementen und Randelementen untersucht. Dabei wird die Rolle des Poincaré-Steklov Operators und seiner Diskretisierung hervorgehoben, die für a priori und a posteriori Fehlerabschätzungen analysiert wird. Die a posteriori Fehlerabschätzungen aus [9, 10] werden verbessert mit Argumenten aus [6] und [9, 10]. Für quasiuniforme Randnetze können mit [5] und [36] Abschätzungen in der umgekehrten Richtung bewiesen werden. Dieses und numerische Beispiele zeigen die Effizienz der Abschätzung.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Babuŝka, I., Miller, A.: A feedback finite element method with a posteriori error estimation. Comp. Meth. Appl. Mech. Eng.61, 1–40 (1987).
Carstensen, C.: Interface problem in holonomic elastoplasticity. Math. Meth. Appl. Sci.16, 819–835 (1993).
Carstensen, C.: Coupling of FEM and BEM for interface problems in viscoplasticity and plasticity with hardening. SIAM J. Numer. Anal.33, 171–206 (1996).
Carstensen, C.: Adaptive boundary element methods and adaptive finite element and boundary element coupling. In: Boundary value problems and integral equations on non-smooth domains (Costabel, M., Dauge, M., Nicaise, S., eds.), pp. 47–58. New York: Marcel Dekker.
Carstensen, C.: Efficiency of a posteriori BEM-error estimates for first kind integral equations on quasi-uniform meshes. Math. Comp.65, 69–84 (1996).
Carstensen, C.: An a posteriori error estimate for a first kind integral equation; Math. Comp. accepted for publication (1996).
Carstensen, C., Stephan, E. P.: A posteriori error estimates for boundary element methods. Math. Comp.64, 483–500 (1994).
Carstensen, C., Stephan, E. P.: Adaptive boundary element methods for some first kind integral equations. SIAM J. Numer. Anal. accepted for publication (1995).
Carstensen, C., Stephan, E. P.: Adaptive coupling of boundary elements and finite elements. Math. Modelling Numer. Anal.29, 1–39 (1995).
Carstensen, C., Funken, S., Stephan, E. P.: On the adaptive coupling of FEM and BEM in 2-d-elasticity. Numer. Math. accepted for publication (1996).
Carstensen, C. Stephan, E. P.: Coupling of FEM and BEM for a nonlinear interface problem: the h-p Version. Numer. Meth. Partial Diff. Eq.111, 539–554 (1995).
Clement, P.: Approximation by finite element functions using local regularization. RAIRO Sér. Rouge Anal. Numér.R-2, 77–84 (1975).
Costabel, M.: Symmetric methods for the coupling of finite elements and boundary elements. In: Boundary elements IX, Vol. 1, (Bretia, C. A., ed.), pp. 411–420 Heidelberg New York Tokyo: Springer 1987.
Costabel, M.: Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal.19, 613–626 (1988).
Costabel, M., Stephan, E. P.: Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation. Banach Center Publ.15, 175–251 (1985).
Costabel, M., Stephan, E. P.: A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl.106, 367–413 (1985).
Costabel, M., Stephan, E. P.: Coupling of finite and boundary element methods for an elastoplastic interface problem. SIAM J. Numer. Anal.27, 1212–1226 (1990).
Eriksson, K., Johnson, C.: An adaptive finite element method for linear elliptic problems. Math. Comp.50, 361–3883 (1988).
Eriksson, K., Johsnon, C.: Adaptive finite element methods for parabolic problems I. A linear model problem. SIAM J. Numer. Anal.28, 43–77 (1991).
Faermann, B.: Lokale a-posteriori-Fehlerschätzer bei der Diskretisierung von Randintegralgleichungen. PhD-thesis, University of Kiel, FRG (1993).
Gaier, D.: Integralgleichungen erster Art und konforme Abbildung, Math. Z.147, 113–129 (1976).
Gatica, G. N., Hsiao, G. C.: On a class of variational formulations for some nonlinear interface problems. Rendiconti di Mathematica Ser. VII10, 681–715 (1990).
Gatica, G. N., Hsiao, G. C.: On the coupled BEM and FEM for a nonlinear exterior Dirichlet problem inR 2. Numer. Math.61, 171–214 (1992).
Han, H.: A new class of variational formulations for the coupling of finite and boundary element methods. J. Comput. Math.8, 223–232 (1990).
Heuer, N.: hp-Versionen der Randelementemethode. PhD Dissertation, University of Hannover, (1992).
Hsiao, G. C., Wendland, W. L.: The Aubin-Nitsche lemma for integral equations. J Integr. Eq.3, 299–315 (1981).
Johnson, C., Hansbo, P.: Adaptive finite element methods in computational mechanics. Comput. Meth. Appl. Mech. Engin.101, 143–181 (1992).
Lions, J. L., Magenes, E.: Non-homogeneous boundary value problems and applications, Vol. I. Berlin Heidelberg New York: Springer 1972.
von Petersdorff, T.: Randwertprobleme der Elastizitätstheorie für Polyeder—Singularitäten und Approximation mit Randelementmethoden. PhD-thesis, TH Darmstadt FRG (1989).
Rank, E.: Adaptive boundary element methods. In: Boundary elements9 (Brebbia, C. A., Wendland, W. L., Kuhn, G., eds.), Vol. 1, 259–273, Berlin Heidelberg New York Tokyo: Springer 1987.
Saranen, J., Wendland, W. L.: Local residual-type error estimates for adaptive boundary element methods on closed curves. Appl. Anal.48, 37–50 (1993).
Sloan, I. H., Spence, A.: The Galerkin method for integral equations of the first kind with logarithmic kernel: theory. IMA J. Numer. Anal.8, 105–122 (1988).
Stephan, E. P., Suri, M.: The hp-version of the boundary element method on polygonal domains with quasiuniform meshes. Math. Modell. Numer. Anal.25, 783–807 (1991).
Stephan, E. P., Wendland W. L.: Remarks on Galerkin and least squares methods with finite elements for general elliptic problems. Manusc. Geod.1, 93–123 (1976).
Stephan, E. P., Wendland, W. L., Hsiao, G. C.: On the integral equation method for the plane mixed boundary value problem of the Laplacian. Math. Meth. Appl. Sci.1, 265–321 (1979).
Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner Skripten zur Numerik. Stuttgart: B.G. Teubner 1996.
Wendland, W. L.: On asymptotic error estimates for combined BEM and FEM. In: Finite and boundary element techniques from mathematical and engineering point of view (Stein, E., Wendland, W. L., eds.), pp. 273–331. New York: Springer 1988.
Wendland, W. L., Yu, D.: Adaptive boundary element methods for strongly elliptic integral equations. Numer. Math.53, 539–558 (1988).
Wendland, W. L., Yu, D.: A posteriori local error estimates of boundary element methods with some pseudo-differential equations on closed curves. J. Comput. Math.10, 273–289 (1992).
Author information
Authors and Affiliations
Additional information
Dedicated to Professor Dr.-Ing. Wolfgang Wendland on the occasion of his 60th birthday
Rights and permissions
About this article
Cite this article
Carstensen, C. A posteriori error estimate for the symmetric coupling of finite elements and boundary elements. Computing 57, 301–322 (1996). https://doi.org/10.1007/BF02252251
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02252251