Boundary element methods for potential problems with nonlinear boundary conditions
Authors:
M. Ganesh and O. Steinbach
Journal:
Math. Comp. 70 (2001), 10311042
MSC (2000):
Primary 31C20, 65L20, 65N38, 74S15
Published electronically:
June 12, 2000
MathSciNet review:
1826575
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Galerkin boundary element methods for the solution of novel first kind SteklovPoincaré and hypersingular operator boundary integral equations with nonlinear perturbations are investigated to solve potential type problems in two and threedimensional Lipschitz domains with nonlinear boundary conditions. For the numerical solution of the resulting Newton iterate linear boundary integral equations, we propose practical variants of the Galerkin scheme and give corresponding error estimates. We also discuss the actual implementation process with suitable preconditioners and propose an optimal hybrid solution strategy.
 1.
Kendall
E. Atkinson and Graeme
Chandler, Boundary integral equation methods for
solving Laplace’s equation with nonlinear boundary conditions: the
smooth boundary case, Math. Comp.
55 (1990), no. 192, 451–472. MR 1035924
(91d:65181), http://dx.doi.org/10.1090/S0025571819901035924X
 2.
Owe
Axelsson, Iterative solution methods, Cambridge University
Press, Cambridge, 1994. MR 1276069
(95f:65005)
 3.
R. Bialecki, A. J. Nowak, Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions. Appl. Math. Model. 5 (1981) 417421.
 4.
C.
Carstensen, M.
Kuhn, and U.
Langer, Fast parallel solvers for symmetric boundary element domain
decomposition equations, Numer. Math. 79 (1998),
no. 3, 321–347. MR 1626312
(99d:65336), http://dx.doi.org/10.1007/s002110050342
 5.
Philippe
G. Ciarlet, The finite element method for elliptic problems,
NorthHolland Publishing Co., AmsterdamNew YorkOxford, 1978. Studies in
Mathematics and its Applications, Vol. 4. MR 0520174
(58 #25001)
 6.
Martin
Costabel, Boundary integral operators on Lipschitz domains:
elementary results, SIAM J. Math. Anal. 19 (1988),
no. 3, 613–626. MR 937473
(89h:35090), http://dx.doi.org/10.1137/0519043
 7.
Robert
L. Doucette, A collocation method for the numerical solution of
Laplace’s equation with nonlinear boundary conditions on a
polygon, SIAM J. Numer. Anal. 30 (1993), no. 3,
717–732. MR 1220648
(94f:65113), http://dx.doi.org/10.1137/0730035
 8.
C. Eck, O. Steinbach, W. L. Wendland, A symmetric boundary element method for contact problems with friction. Math. Comput. Simulation 50 (1999) 4159. CMP 2000:03
 9.
P.
P. B. Eggermont and J.
Saranen, 𝐿^{𝑝} estimates of boundary integral
equations for some nonlinear boundary value problems, Numer. Math.
58 (1990), no. 5, 465–478. MR 1080302
(91m:65325), http://dx.doi.org/10.1007/BF01385636
 10.
M.
Ganesh, A BIE method for a nonlinear BVP, J. Comput. Appl.
Math. 45 (1993), no. 3, 299–308. MR 1216073
(94f:65116), http://dx.doi.org/10.1016/03770427(93)90047F
 11.
M.
Ganesh, I.
G. Graham, and J.
Sivaloganathan, A pseudospectral threedimensional boundary
integral method applied to a nonlinear model problem from finite
elasticity, SIAM J. Numer. Anal. 31 (1994),
no. 5, 1378–1414. MR 1293521
(95g:45011), http://dx.doi.org/10.1137/0731072
 12.
M. Ganesh, O. Steinbach, Nonlinear boundary integral equations for harmonic problems. J. Int. Equations. Appl. 11(4) (1999).
 13.
G. C. Hsiao, W. L. Wendland, The AubinNitsche Lemma for integral equations. J. Int. Equations. Appl. 3 (1981) 299315.
 14.
F. Incropera, D. DeWitt, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, New York, 1990.
 15.
D.
B. Ingham, P.
J. Heggs, and M.
Manzoor, Boundary integral equation solution of nonlinear plane
potential problems, IMA J. Numer. Anal. 1 (1981),
no. 4, 415–426. MR 641319
(83a:65122), http://dx.doi.org/10.1093/imanum/1.4.415
 16.
M. A. Kelmanson, Solution of nonlinear elliptic equations with boundary singularities by an integral equation method. J. Comp. Phys. 56 (1984) 244258.
 17.
K.
Ruotsalainen and J.
Saranen, On the collocation method for a nonlinear boundary
integral equation, Proceedings of the 3rd International Congress on
Computational and Applied Mathematics (Leuven, 1988), 1989,
pp. 339–348. MR 1038855
(91d:65180), http://dx.doi.org/10.1016/03770427(89)903452
 18.
K.
Ruotsalainen and W.
Wendland, On the boundary element method for some nonlinear
boundary value problems, Numer. Math. 53 (1988),
no. 3, 299–314. MR 948589
(89h:65189), http://dx.doi.org/10.1007/BF01404466
 19.
Y. Saad, Iterative Methods for Sparse Linear Systems. PWS, Boston, 1996.
 20.
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 (92f:65004)
 21.
O.
Steinbach and W.
L. Wendland, The construction of some efficient preconditioners in
the boundary element method, Adv. Comput. Math. 9
(1998), no. 12, 191–216. Numerical treatment of boundary
integral equations. MR 1662766
(99j:65219), http://dx.doi.org/10.1023/A:1018937506719
 1.
 K. E. Atkinson, G. A. Chandler, BIE methods for solving Laplace's equation with nonlinear boundary conditions. Math. Comp. 55 (1990) 451457. MR 91d:65181
 2.
 O. Axelsson, Iterative Solution Methods. Cambridge University Press, 1994. MR 95f:65005
 3.
 R. Bialecki, A. J. Nowak, Boundary value problems in heat conduction with nonlinear material and nonlinear boundary conditions. Appl. Math. Model. 5 (1981) 417421.
 4.
 C. Carstensen, M. Kuhn, U. Langer, Fast parallel solvers for symmetric boundary element domain decomposition equations. Numer. Math. 79 (1998) 321347. MR 99d:65336
 5.
 P. G. Ciarlet, The Finite Element Method for Elliptic Problems. NorthHolland, 1978. MR 58:25001
 6.
 M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results. SIAM J. Math. Anal. 19 (1988) 613626. MR 89h:35090
 7.
 R. L. Doucette, A collocation method for the numerical solution of Laplace's equation with nonlinear boundary conditions on a polygon. SIAM J. Numer. Anal. 30 (1993) 717732. MR 94f:65113
 8.
 C. Eck, O. Steinbach, W. L. Wendland, A symmetric boundary element method for contact problems with friction. Math. Comput. Simulation 50 (1999) 4159. CMP 2000:03
 9.
 P. P. B. Eggermont, J. Saranen, estimates of boundary integral equations for some nonlinear boundary value problems. Numer. Math. 58 (1990) 465478. MR 91m:65325
 10.
 M. Ganesh, A BIE method for a nonlinear BVP. J. Comput. Appl. Math. 45 (1993) 299308. MR 94f:65116
 11.
 M. Ganesh, I. G. Graham and J. Sivaloganathan, A pseudospectral threedimensional boundary integral method applied to a nonlinear model problems from finite elasticity. SIAM J. Numer. Anal. 31 (1994) 13781414. MR 95g:45011
 12.
 M. Ganesh, O. Steinbach, Nonlinear boundary integral equations for harmonic problems. J. Int. Equations. Appl. 11(4) (1999).
 13.
 G. C. Hsiao, W. L. Wendland, The AubinNitsche Lemma for integral equations. J. Int. Equations. Appl. 3 (1981) 299315.
 14.
 F. Incropera, D. DeWitt, Fundamentals of Heat and Mass Transfer, John Wiley & Sons, New York, 1990.
 15.
 D. B. Ingham, P. J. Heggs, M. Manzoor, Boundary integral equation solution of nonlinear plane potential problem. IMA J. Numer. Anal. 1 (1981) 415426. MR 83a:65122
 16.
 M. A. Kelmanson, Solution of nonlinear elliptic equations with boundary singularities by an integral equation method. J. Comp. Phys. 56 (1984) 244258.
 17.
 K. Ruotsalainen, J. Saranen, On the collocation method for a nonlinear boundary integral equation. J. Comput. Appl. Math. 28 (1989) 339348. MR 91d:65180
 18.
 K. Ruotsalainen, W. L. Wendland, On the boundary element method for some nonlinear boundary value problems. Numer. Math. 53 (1988) 299314. MR 89h:65189
 19.
 Y. Saad, Iterative Methods for Sparse Linear Systems. PWS, Boston, 1996.
 20.
 A. H. Schatz, V. Thomée, W. L. Wendland, Mathematical Theory of Finite and Boundary Element Methods. Birkhäuser, Basel, 1990. MR 92f:65004
 21.
 O. Steinbach, W. L. Wendland, The construction of some efficient preconditioners in the boundary element method. Adv. Comput. Math. 9 (1998) 191216. MR 99j:65219
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC (2000):
31C20,
65L20,
65N38,
74S15
Retrieve articles in all journals
with MSC (2000):
31C20,
65L20,
65N38,
74S15
Additional Information
M. Ganesh
Affiliation:
School of Mathematics, University of New South Wales, Sydney 2052, Australia
Email:
ganesh@maths.unsw.edu.au
O. Steinbach
Affiliation:
Mathematisches Institut A, Universität Stuttgart, Pfaffenwaldring 57, 70569 Stuttgart, Germany
Email:
steinbach@mathematik.unistuttgart.de
DOI:
http://dx.doi.org/10.1090/S0025571800012667
PII:
S 00255718(00)012667
Keywords:
Boundary element methods,
Nonlinear boundary conditions
Received by editor(s):
September 10, 1998
Received by editor(s) in revised form:
November 3, 1998, and July 30, 1999
Published electronically:
June 12, 2000
Additional Notes:
Part of this work was carried out while the second author was a Visiting Fellow in the School of Mathematics, UNSW, under an Australian Research Council Grant. The support of the Australian Research Council is gratefully acknowledged by both authors.
Dedicated:
Dedicated to Professor Ian Sloan on the occasion of his 60th birthday
Article copyright:
© Copyright 2000
American Mathematical Society
