Optimal order collocation for the mixed boundary value problem on polygons

Laubin, Pascal

doi:10.1090/S0025-5718-00-01209-6

Optimal order collocation for the mixed boundary value problem on polygons
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Math. Comp. 70 (2001), 607-636 Request permission

Abstract:

In usual boundary elements methods, the mixed Dirichlet-Neumann problem in a plane polygonal domain leads to difficulties because of the transition of spaces in which the problem is well posed. We build collocation methods based on a mixed single and double layer potential. This indirect method is constructed in such a way that strong ellipticity is obtained in high order spaces of Sobolev type. The boundary values of this potential define a bijective boundary operator if a modified capacity adapted to the problem is not $1$. This condition is analogous to the one met in the use of the single layer potential, and is not a problem in practical computations. The collocation methods use smoothest splines and known singular functions generated by the corners. If splines of order $2m-1$ are used, we get quasi-optimal estimates in $H^m$-norm. The order of convergence is optimal in the sense that it is fixed by the approximation properties of the first missed singular function.

References

Carlos A. Berenstein and Roger Gay, Complex variables, Graduate Texts in Mathematics, vol. 125, Springer-Verlag, New York, 1991. An introduction. MR 1107514, DOI 10.1007/978-1-4612-3024-3
Martin Costabel, Boundary integral operators on Lipschitz domains: elementary results, SIAM J. Math. Anal. 19 (1988), no. 3, 613–626. MR 937473, DOI 10.1137/0519043
Martin Costabel and Ernst Stephan, Boundary integral equations for mixed boundary value problems in polygonal domains and Galerkin approximation, Mathematical models and methods in mechanics, Banach Center Publ., vol. 15, PWN, Warsaw, 1985, pp. 175–251. MR 874845
Martin Costabel and Ernst P. Stephan, On the convergence of collocation methods for boundary integral equations on polygons, Math. Comp. 49 (1987), no. 180, 461–478. MR 906182, DOI 10.1090/S0025-5718-1987-0906182-9
Martin Costabel and Ernst P. Stephan, Duality estimates for the numerical solution of integral equations, Numer. Math. 54 (1988), no. 3, 339–353. MR 971707, DOI 10.1007/BF01396766
Robert Dautray and Jacques-Louis Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 4, INSTN: Collection Enseignement. [INSTN: Teaching Collection], Masson, Paris, 1988 (French). Méthodes variationnelles. [Variational methods]; With the collaboration of Michel Artola, Marc Authier, Michel Cessenat, Jean Michel Combes, Bertrand Mercier and Claude Wild; Reprinted from the 1984 edition. MR 944302
J. Elschner and I. G. Graham, An optimal order collocation method for first kind boundary integral equations on polygons, Numer. Math. 70 (1995), no. 1, 1–31. MR 1320699, DOI 10.1007/s002110050107
J. Elschner, Y. Jeon, I. H. Sloan, and E. P. Stephan, The collocation method for mixed boundary value problems on domains with curved polygonal boundaries, Numer. Math. 76 (1997), no. 3, 355–381. MR 1452513, DOI 10.1007/s002110050267
Dieter Gaier, Integralgleichungen erster Art und konforme Abbildung, Math. Z. 147 (1976), no. 2, 113–129. MR 396926, DOI 10.1007/BF01164277
P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
C. J. Everett Jr., Annihilator ideals and representation iteration for abstract rings, Duke Math. J. 5 (1939), 623–627. MR 13
David S. Jerison and Carlos E. Kenig, The Dirichlet problem in nonsmooth domains, Ann. of Math. (2) 113 (1981), no. 2, 367–382. MR 607897, DOI 10.2307/2006988
David S. Jerison and Carlos E. Kenig, The Neumann problem on Lipschitz domains, Bull. Amer. Math. Soc. (N.S.) 4 (1981), no. 2, 203–207. MR 598688, DOI 10.1090/S0273-0979-1981-14884-9
Pascal Laubin, High order convergence for collocation of second kind boundary integral equations on polygons, Numer. Math. 79 (1998), no. 1, 107–140. MR 1608421, DOI 10.1007/s002110050333
Pascal Laubin and Marc Baiwir, Spline collocation for a boundary integral equation on polygons with cuts, SIAM J. Numer. Anal. 35 (1998), no. 4, 1452–1472. MR 1620093, DOI 10.1137/S0036142997318358
Hidegorô Nakano, Über Abelsche Ringe von Projektionsoperatoren, Proc. Phys.-Math. Soc. Japan (3) 21 (1939), 357–375 (German). MR 94
Bert-Wolfgang Schulze, Pseudo-differential boundary value problems, conical singularities, and asymptotics, Mathematical Topics, vol. 4, Akademie Verlag, Berlin, 1994. MR 1282496
Gregory Verchota, Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains, J. Funct. Anal. 59 (1984), no. 3, 572–611. MR 769382, DOI 10.1016/0022-1236(84)90066-1
W. L. Wendland, E. Stephan, and G. C. Hsiao, On the integral equation method for the plane mixed boundary value problem of the Laplacian, Math. Methods Appl. Sci. 1 (1979), no. 3, 265–321. MR 548943, DOI 10.1002/mma.1670010302
Albert H. Schatz, Vidar Thomée, and Wolfgang L. Wendland, Mathematical theory of finite and boundary element methods, DMV Seminar, vol. 15, Birkhäuser Verlag, Basel, 1990. MR 1116555, DOI 10.1007/978-3-0348-7630-8