# Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.98.

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## On the coupling of boundary integral and finite element methodsHTML articles powered by AMS MathViewer

by Claes Johnson and J.-Claude Nédélec
Math. Comp. 35 (1980), 1063-1079 Request permission

## Abstract:

Let ${\Omega ^c}$ be the complementary of a bounded regular domain in ${{\mathbf {R}}^2}$ of boundary $\Gamma$. We consider the problem (1) $\left \{ {\begin {array}{*{20}{c}} {\Delta u = f;} & {{\text {in}}\;{\Omega ^c},} \\ {u{|_\Gamma } = {u_{0,}}} & {} \\ \end {array} } \right .$ 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) $\left \{ {\begin {array}{*{20}{c}} {\Delta u = f} \hfill & {{\text {in}}\;{\Omega _1},} \hfill \\ {u{|_\Gamma } = {u_0},} \hfill & {} \hfill \\ {u{|_{{\Gamma _2}}} = \lambda .} \hfill & {} \hfill \\ \end {array} } \right .$ We introduce a finite element approximation of Eq. (2) and of the integral equation and we prove optimal error estimates.
References
• Philippe G. Ciarlet, The finite element method for elliptic problems, Studies in Mathematics and its Applications, Vol. 4, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. MR 0520174
• M. DJAOUA, Equations Intégrales pour un Problème Singulier dans le Plan, These 3é cycle, Paris, 1977. J. GIROIRE, Rapport Interne à l’Ecole Polytechnique, Centre de Mathématiques Appliquées, Palaiseau, 1976.
• Donald Greenspan and Peter Werner, A numerical method for the exterior Dirichlet problem for the reduced wave equation, Arch. Rational Mech. Anal. 23 (1966), 288–316. MR 238501, DOI 10.1007/BF00281165
• George C. Hsiao and Wolfgang L. Wendland, A finite element method for some integral equations of the first kind, J. Math. Anal. Appl. 58 (1977), no. 3, 449–481. MR 461963, DOI 10.1016/0022-247X(77)90186-X
• M. N. Le Roux, Méthode d’éléments finis pour la résolution numérique de problèmes extérieurs en dimension $2$, RAIRO Anal. Numér. 11 (1977), no. 1, 27–60, 112 (French, with English summary). MR 448954, DOI 10.1051/m2an/1977110100271
• J. C. NEDELEC, Cours de l’Ecole d’Eté d’Analyse Numérique, C.E.A., I.R.I.A., E.P.F., 1977.
• J.-C. Nédélec and J. Planchard, Une méthode variationnelle d’éléments finis pour la résolution numérique d’un problème extérieur dans $R^{3}$, Rev. Française Automat. Informat. Recherche Opérationnelle Sér. Rouge 7 (1973), no. R-3, 105–129 (French, with English summary). MR 424022
• R. Seeley, Topics in pseudo-differential operators, Pseudo-Diff. Operators (C.I.M.E., Stresa, 1968) Edizioni Cremonese, Rome, 1969, pp. 167–305. MR 0259335
• P. Silvester and M.-S. Hsieh, Finite-element solution of $2$-dimensional exterior-field problems, Proc. Inst. Electr. Engrs. 118 (1971), 1743–1748. MR 334553, DOI 10.1049/piee.1971.0320
• O. C. Zienkiewicz, D. W. Kelly, and P. Bettess, The coupling of the finite element method and boundary solution procedures, Internat. J. Numer. Methods Engrg. 11 (1977), no. 2, 355–375. MR 451784, DOI 10.1002/nme.1620110210
• W. L. Wendland, Elliptic systems in the plane, Monographs and Studies in Mathematics, vol. 3, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. MR 518816
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