Approximation methods for nonlinear problems with application to twopoint boundary value problems
Author:
H. B. Keller
Journal:
Math. Comp. 29 (1975), 464474
MSC:
Primary 65J05; Secondary 65L05
MathSciNet review:
0371058
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: General nonlinear problems in the abstract form and corresponding families of approximating problems in the form are considered (in an appropriate Banach space setting). The relation between "isolation" and "stability" of solutions is briefly studied. The main result shows, essentially, that, if the nonlinear problem has an isolated solution and the approximating family has stable Lipschitz continuous linearizations, then the approximating problem has a stable solution which is close to the exact solution. Error estimates are obtained and Newton's method is shown to converge quadratically. These results are then used to justify a broad class of difference schemes (resembling linear multistep methods) for general nonlinear twopoint boundary value problems.
 [1]
Herbert
B. Keller, Accurate difference methods for linear ordinary
differential systems subject to linear constraints, SIAM J. Numer.
Anal. 6 (1969), 8–30. MR 0253562
(40 #6776)
 [2]
Herbert
B. Keller, Newton’s method under mild differentiability
conditions, J. Comput. System Sci. 4 (1970),
15–28. MR
0250476 (40 #3710)
 [3]
Herbert
B. Keller, Accurate difference methods for nonlinear twopoint
boundary value problems, SIAM J. Numer. Anal. 11
(1974), 305–320. MR 0351098
(50 #3589)
 [4]
H.
B. Keller and A.
B. White Jr., Difference methods for boundary value problems in
ordinary differential equations, SIAM J. Numer. Anal.
12 (1975), no. 5, 791–802. MR 0413513
(54 #1627)
 [5]
HeinzOtto
Kreiss, Difference approximations for boundary
and eigenvalue problems for ordinary differential equations, Math. Comp. 26 (1972), 605–624. MR 0373296
(51 #9496), http://dx.doi.org/10.1090/S00255718197203732963
 [6]
J.
M. Ortega and W.
C. Rheinboldt, Iterative solution of nonlinear equations in several
variables, Academic Press, New YorkLondon, 1970. MR 0273810
(42 #8686)
 [7]
Robert
D. Richtmyer, Difference methods for initialvalue problems,
Interscience tracts in pure and applied mathematics. Iract 4, Interscience
Publishers, Inc., New. York, 1957. MR 0093918
(20 #438)
 [8]
R.
B. Simpson, Existence and error estimates for
solutions of a discrete analog of nonlinear eigenvalue problems,
Math. Comp. 26 (1972), 359–375. MR 0315918
(47 #4466), http://dx.doi.org/10.1090/S00255718197203159189
 [9]
Hans
J. Stetter, Asymptotic expansions for the error of discretization
algorithms for nonlinear functional equations, Numer. Math.
7 (1965), 18–31. MR 0175320
(30 #5505)
 [10]
Hans
J. Stetter, Stability of nonlinear discretization algorithms,
Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ.
Maryland, 1965) Academic Press, New York, 1966, pp. 111–123.
MR
0205495 (34 #5322)
 [11]
Richard
Weiss, The application of implicit
RungeKutta and collection methods to boundaryvalue problems, Math. Comp. 28 (1974), 449–464. MR 0341881
(49 #6627), http://dx.doi.org/10.1090/S00255718197403418812
 [12]
A. B. WHITE, Numerical Solution of Two Point Boundary Value Problems, Ph. D. Thesis, Calif. Inst. of Technology, Pasadena, 1974.
 [13]
Victor
Pereyra, Iterated deferred corrections for nonlinear operator
equations, Numer. Math. 10 (1967), 316–323. MR 0221760
(36 #4812)
 [1]
 H. B. KELLER, "Accurate difference methods for linear ordinary differential systems subject to linear constraints," SIAM J. Numer. Anal., v. 6, 1969, pp. 830. MR 40 #6776. MR 0253562 (40:6776)
 [2]
 H. B. KELLER, "Newton's method under mild differentiability conditions," J. Comput. System Sci., v. 4, 1970, pp. 1528. MR 40 #3710. MR 0250476 (40:3710)
 [3]
 H. B. KELLER, "Accurate difference methods for nonlinear two point boundary value problems," SIAM J. Numer. Anal., v. 11, 1974, pp. 305320. MR 0351098 (50:3589)
 [4]
 H. B. KELLER & A. B. WHITE, "Difference methods for boundary value problems in ordinary differential equations," SIAM J. Numer. Anal., v. 12, 1975. (To appear.) MR 0413513 (54:1627)
 [5]
 H.O. KREISS, "Difference approximations for boundary and eigenvalue problems for ordinary differential equations," Math. Comp., v. 26, 1972, pp. 605624. MR 0373296 (51:9496)
 [6]
 J. M. ORTEGA & W. C. RHEINBOLDT, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970. MR 42 #8686. MR 0273810 (42:8686)
 [7]
 R. D. RICHTMYER, Difference Methods for InitialValue Problems, Interscience Tracts in Pure and Appl. Math., no. 4, Interscience, New York, 1957. MR 20 #438. MR 0093918 (20:438)
 [8]
 R. B. SIMPSON, "Existence and error estimates for solutions of a discrete analog of nonlinear eigenvalue problems," Math. Comp., v. 26, 1972, pp. 359375. MR 47 #4466. MR 0315918 (47:4466)
 [9]
 H. J. STETTER, "Asymptotic expansions for the error of discretization algorithms for nonlinear functional equations," Numer. Math., v. 7, 1965, pp. 1831. MR 30 #5505. MR 0175320 (30:5505)
 [10]
 H. J. STETTER, "Stability of nonlinear discretization algorithms," Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965), Academic Press, New York, 1966, pp. 111123. MR 34 #5322. MR 0205495 (34:5322)
 [11]
 R. K. WEISS, "The application of implicit RungeKutta and collocation methods to boundary value problems," Math. Comp., v. 28, 1974, pp. 449464. MR 0341881 (49:6627)
 [12]
 A. B. WHITE, Numerical Solution of Two Point Boundary Value Problems, Ph. D. Thesis, Calif. Inst. of Technology, Pasadena, 1974.
 [13]
 V. PEREYRA, "Iterated deferred corrections for nonlinear operator equations," Numer. Math., v. 10, 1967, pp. 316323. MR 0221760 (36:4812)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65J05,
65L05
Retrieve articles in all journals
with MSC:
65J05,
65L05
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197503710587
PII:
S 00255718(1975)03710587
Keywords:
Nonlinear stability,
Newton's method,
two point boundary problems,
finite difference schemes
Article copyright:
© Copyright 1975
American Mathematical Society
