Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

On the coupling of boundary integral and finite element methods


Authors: Claes Johnson and J.-Claude Nédélec
Journal: Math. Comp. 35 (1980), 1063-1079
MSC: Primary 65N30
DOI: https://doi.org/10.1090/S0025-5718-1980-0583487-9
MathSciNet review: 583487
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ {\Omega ^c}$ be the complementary of a bounded regular domain in $ {{\mathbf{R}}^2}$ of boundary $ \Gamma $. We consider the problem (1)

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\Delta u = f;} & {{\text{in}}\;{\Omega ^c},} \\ {u{\vert _\Gamma } = {u_{0,}}} & {} \\ \end{array} } \right.$

where f has its support in a bounded subdomain $ {\Omega _1}$ of $ {\Omega ^c}$. Let $ {\Gamma _2}$ be the common boundary of $ {\Omega _1}$ and $ {\Omega _2} = {\Omega ^c} - {\Omega _1}$. We solve the problem (1) by using an equivalent system of equations involving an integral equation on $ {\Gamma ^2}$ coupled with the equation: (2)

$\displaystyle \left\{ {\begin{array}{*{20}{c}} {\Delta u = f} \hfill & {{\text{... ...{\vert _{{\Gamma _2}}} = \lambda .} \hfill & {} \hfill \\ \end{array} } \right.$

We introduce a finite element approximation of Eq. (2) and of the integral equation and we prove optimal error estimates.

References [Enhancements On Off] (What's this?)

  • [1] P. G. CIARLET, The Finite Element Method for Elliptic Problems, North-Holland, Amsterdam, 1978. MR 0520174 (58:25001)
  • [2] M. DJAOUA, Equations Intégrales pour un Problème Singulier dans le Plan, These 3é cycle, Paris, 1977.
  • [3] J. GIROIRE, Rapport Interne à l'Ecole Polytechnique, Centre de Mathématiques Appliquées, Palaiseau, 1976.
  • [4] D. GREENSPAN & P. WERNER, "A numerical method for the exterior Dirichlet problem for the reduced wave equation," Arch. Rational Mech. Anal., v. 23, 1966/67, pp. 288-316. MR 0238501 (38:6777)
  • [5] G. C. HSIAO & W. L. WENDLAND, "A finite element method for some integral equations of the first kind," J. Math. Anal. Appl., v. 58, 1977, pp. 449-481. MR 0461963 (57:1945)
  • [6] M. N. LE ROUX, "Méthode d'éléments finis pour la résolution numérique de problèmes extérieurs en dimension 2," R.A.I.R.O. Anal. Numér., v. 11, 1977, pp. 27-60. MR 0448954 (56:7259)
  • [7] J. C. NEDELEC, Cours de l'Ecole d'Eté d'Analyse Numérique, C.E.A., I.R.I.A., E.P.F., 1977.
  • [8] J. C. NEDELEC & J. PLANCHARD, "Une méthode variationelle d'éléments finis pour la résolution numérique d'un problème extérieur dans $ {{\mathbf{R}}^3}$," R.A.I.R.O., v. 7, 1973, pp. 105-129. MR 0424022 (54:11992)
  • [9] R. SEELEY, Cours du C.I.M.E. de 1968, ed. Cremonese, Roma, 1969. MR 0259335 (41:3973)
  • [10] P. SILVESTER & M. S. HSIEH, "Finite element solution of 2-dimensional exterior field problems," Proc. Inst. Electr. Engrs., v. 118, 1971, pp. 1743-1746. MR 0334553 (48:12872)
  • [11] O. C. ZIENKIEWICZ, D. W. KELLY & P. BETTESS, "The coupling of the finite element method and boundary solution procedures," Internat. J. Numer. Methods Engrg., v. 11, 1977, pp. 355-375. MR 0451784 (56:10066)
  • [12] W. L. WENDLAND, Elliptic Systems in the Plane, Pitman, New York, 1979. MR 518816 (80h:35053)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N30

Retrieve articles in all journals with MSC: 65N30


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1980-0583487-9
Article copyright: © Copyright 1980 American Mathematical Society

American Mathematical Society