Boundary element monotone iteration scheme for semilinear elliptic partial differential equations
Authors:
Yuanhua Deng, Goong Chen, WeiMing Ni and Jianxin Zhou
Journal:
Math. Comp. 65 (1996), 943982
MSC (1991):
Primary 31B20, 35J65, 65N38
MathSciNet review:
1348042
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: The monotone iteration scheme is a constructive method for solving a wide class of semilinear elliptic boundary value problems. With the availability of a supersolution and a subsolution, the iterates converge monotonically to one or two solutions of the nonlinear PDE. However, the rates of such monotone convergence cannot be determined in general. In addition, when the monotone iteration scheme is implemented numerically through the boundary element method, error estimates have not been analyzed in earlier studies. In this paper, we formulate a working assumption to obtain an exponentially fast rate of convergence. This allows a margin for the numerical implementation of boundary elements within the range of monotone convergence. We then interrelate several approximate solutions, and use the AubinNitsche lemma and the triangle inequalities to derive error estimates for the Galerkin boundaryelement iterates with respect to the , , Sobolev space norms. Such estimates are of optimal order. Furthermore, as a peculiarity, we show that for the nonlinearities that are of separable type, ``higher than optimal order'' error estimates can be obtained with respect to the mesh parameter . Several examples of semilinear elliptic partial differential equations featuring different situations of existence/nonexistence, uniqueness/multiplicity and stability are discussed, computed, and the graphics of their numerical solutions are illustrated.
 1.
Handbook of mathematical functions, with formulas, graphs and
mathematical tables, Edited by Milton Abramowitz and Irene A. Stegun.
Fifth printing, with corrections. National Bureau of Standards Applied
Mathematics Series, Vol. 55, National Bureau of Standards, Washington,
D.C., (for sale by the Superintendent of Documents, U.S. Government
Printing Office, Washington, D.C., 20402), 1966. MR 0208798
(34 #8607)
 2.
Herbert
Amann, Supersolutions, monotone iterations, and stability, J.
Differential Equations 21 (1976), no. 2,
363–377. MR 0407451
(53 #11226)
 3.
Antonio
Ambrosetti and Paul
H. Rabinowitz, Dual variational methods in critical point theory
and applications, J. Functional Analysis 14 (1973),
349–381. MR 0370183
(51 #6412)
 4.
Ivo
Babuška and A.
K. Aziz, Survey lectures on the mathematical foundations of the
finite element method, The mathematical foundations of the finite
element method with applications to partial differential equations (Proc.
Sympos., Univ. Maryland, Baltimore, Md., 1972) Academic Press, New York,
1972, pp. 1–359. With the collaboration of G. Fix and R. B.
Kellogg. MR
0421106 (54 #9111)
 5.
C.A. Brebbia and S. Walker, Boundary Element Techniques in Engineering, NewnesButterworths, London, 1980.
 6.
Goong
Chen and Jianxin
Zhou, Boundary element methods, Computational Mathematics and
Applications, Academic Press, Ltd., London, 1992. MR 1170348
(93e:65143)
 7.
G. Chen and J. Zhou, Vibration and Damping in Distributed Systems, Vol. II: WKB and Wave Methods, Visualization and Experimentation CRC Press, Boca Raton, Florida, 1993.
 8.
Y. Deng, Boundary element methods for nonlinear partial differential equations,, Ph.D. dissertation, Math. Dept., Texas A&M Univ., College Station, Texas, August 1994.
 9.
Gaetano
Fichera, Linear elliptic equations of higher order in two
independent variables and singular integral equations, with applications to
anistropic inhomogeneous elasticity, Partical differential equations
and continuum mechanics, Univ. of Wisconsin Press, Madison, Wis., 1961,
pp. 55–80. MR 0156084
(27 #6016)
 10.
G.
C. Hsiao and W.
L. Wendland, The AubinNitsche lemma for integral equations,
J. Integral Equations 3 (1981), no. 4, 299–315.
MR 634453
(83j:45019)
 11.
J.L.
Lions, Quelques méthodes de résolution des
problèmes aux limites non linéaires, Dunod;
GauthierVillars, Paris, 1969 (French). MR 0259693
(41 #4326)
 12.
W.M. Ni, Some aspects of semilinear elliptic equations, Lecture Notes published by Institute of Mathematics, National Tsing Hua Univ., Hsinchu, Taiwan, Rep. of China, May, 1987.
 13.
Murray
H. Protter and Hans
F. Weinberger, Maximum principles in differential equations,
PrenticeHall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
(36 #2935)
 14.
K.
Ruotsalainen and J.
Saranen, On the convergence of the Galerkin method for nonsmooth
solutions of integral equations, Numer. Math. 54
(1988), no. 3, 295–302. MR 971704
(90a:65281), http://dx.doi.org/10.1007/BF01396763
 15.
K.
Ruotsalainen and W.
L. Wendland, On the boundary element method for a nonlinear
boundary value problem, Boundary elements IX, Vol.\ 2 (Stuttgart,
1987) Comput. Mech., Southampton, 1987, pp. 385–393. MR 965337
(90e:65161)
 16.
M. Sakakihara, An iterative boundary integral equation method for mildly nonlinear elliptic partial differential equations, Boundary Elements VII (C.A. Brebbia and G. Maier, ed.), vol. . II, SpringerVerlag, BerlinHeidelberg, 1985, pp. 13.4913.58. CMP 20:14
 17.
David
H. Sattinger, Topics in stability and bifurcation theory,
Lecture Notes in Mathematics, Vol. 309, SpringerVerlag, BerlinNew York,
1973. MR
0463624 (57 #3569)
 1.
 M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions, Dover, New York, 1965. MR 34:8607
 2.
 H. Amann, Supersolution, monotone iteration and stability, J. Diff.Eq. 21 (1976), 367377. MR 53:11226
 3.
 A. Ambrosetti and P. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973), 349381. MR 51:6412
 4.
 I. Babu\v{s}ka and A.K. Aziz, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, Academic Press, New York, 1972. MR 54:9111
 5.
 C.A. Brebbia and S. Walker, Boundary Element Techniques in Engineering, NewnesButterworths, London, 1980.
 6.
 G. Chen and J. Zhou, Boundary Element Methods, Academic Press, London, 1992. MR 93e:65143
 7.
 G. Chen and J. Zhou, Vibration and Damping in Distributed Systems, Vol. II: WKB and Wave Methods, Visualization and Experimentation CRC Press, Boca Raton, Florida, 1993.
 8.
 Y. Deng, Boundary element methods for nonlinear partial differential equations,, Ph.D. dissertation, Math. Dept., Texas A&M Univ., College Station, Texas, August 1994.
 9.
 G. Fichera, Linear elliptic equations of higher order in two independent variables and singular integral equations with applications to anisotropic inhomogeneous elasticity, Proc. of Symposium on Partial Differential Equations and Continuum Mechanics ( R.E. Langer, ed.), 5580, Univ. of Wisconsin Press, Madison, Wisconsin, 1961. MR 27:6016
 10.
 G.C. Hsiao and W.L. Wendland, The AubinNitsche lemma for integral equations, J. Integral Eq. 3 (1981), 299315. MR 83j:45019
 11.
 J.L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Lineaires, Dunod, Paris, 1969. MR 41:4326
 12.
 W.M. Ni, Some aspects of semilinear elliptic equations, Lecture Notes published by Institute of Mathematics, National Tsing Hua Univ., Hsinchu, Taiwan, Rep. of China, May, 1987.
 13.
 M. Protter and H. Weinberger, Maximum Principles in Differential Equations, PrenticeHall, Englewood Cliffs, New Jersey, 1967. MR 36:2935
 14.
 K. Ruotsalainen and J. Saranen, On the convergence of the Galerkin method for nonsmooth solutions of integral equations, Num. Math.54 (1988), 295302. MR 90a:65281
 15.
 K. Ruotsalainen and W.L. Wendland, On the boundary element method for a nonlinear boundary value problem, Boundary Elements IX (C.A. Brebbia, ed.), SpringerVerlag, New York, 1987, pp. 385393. MR 90e:65161
 16.
 M. Sakakihara, An iterative boundary integral equation method for mildly nonlinear elliptic partial differential equations, Boundary Elements VII (C.A. Brebbia and G. Maier, ed.), vol. . II, SpringerVerlag, BerlinHeidelberg, 1985, pp. 13.4913.58. CMP 20:14
 17.
 D. Sattinger, Topics in Stability and Bifurcation Theory, Lecture Notes in Mathematics, vol. . 309, SpringerVerlag, New York, 1973. MR 57:3569
Similar Articles
Retrieve articles in Mathematics of Computation of the American Mathematical Society
with MSC (1991):
31B20,
35J65,
65N38
Retrieve articles in all journals
with MSC (1991):
31B20,
35J65,
65N38
Additional Information
Yuanhua Deng
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email:
ydeng@cs.tamu.edu
Goong Chen
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843
Email:
gchen@math.tamu.edu
WeiMing Ni
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843
Jianxin Zhou
Affiliation:
Department of Mathematics, Texas A&M University, College Station, Texas 77843;
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455
Email:
jzhou@math.tamu.edu
DOI:
http://dx.doi.org/10.1090/S0025571896007430
PII:
S 00255718(96)007430
Keywords:
Numerical PDE,
boundary elements,
potential theory,
nonlinear PDE,
elliptic type
Received by editor(s):
May 4, 1994
Received by editor(s) in revised form:
March 1, 1995
Additional Notes:
The first, second, and fourth authors were supported in part by AFOSR Grant 910097 and NSF Grant DMS 9404380.
The third author was supported in part by NSF Grants DMS 9101446 and 9401333.
The second author was on sabbatical leave at the Institute of Applied Mathematics, National Tsing Hua University, Hsinchu, Taiwan, ROC. Supported in part by a grant from the National Science Council of the Republic of China.
Article copyright:
© Copyright 1996
American Mathematical Society
