Existence and error estimates for solutions of a discrete analog of nonlinear eigenvalue problems

Author:
R. B. Simpson

Journal:
Math. Comp. **26** (1972), 359-375

MSC:
Primary 65N25

DOI:
https://doi.org/10.1090/S0025-5718-1972-0315918-9

MathSciNet review:
0315918

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Finite-difference methods using the five-point discrete Laplacian and suitable boundary modifications for approximating in a plane domain on its boundary are considered. It is shown that if (1) has an isolated solution, , then the discrete problem has a solution, , for which . If the discrete problem has solutions, , such that as tends to zero, then (1) has a solution, , satisfying . Let be a critical value of so that (1) has positive solutions for but not for , then the discrete problem has an analogous critical value and, under suitable conditions, . Computed results for the case and the unit square are given.

**[1]**Shmuel Agmon,*Lectures on elliptic boundary value problems*, Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR**0178246****[2]**J. H. Bramble,*On the convergence of difference approximations to weak solutions of Dirichlet’s problem*, Numer. Math.**13**(1969), 101–111. MR**0250495**, https://doi.org/10.1007/BF02163229**[3]**J. H. Bramble and B. E. Hubbard,*On the formulation of finite difference analogues of the Dirichlet problem for Poisson’s equation*, Numer. Math.**4**(1962), 313–327. MR**0149672**, https://doi.org/10.1007/BF01386325**[4]**J. H. Bramble,*Error estimates for difference methods in forced vibration problems*, SIAM J. Numer. Anal.**3**(1966), no. 1, 1–12. MR**0201084**, https://doi.org/10.1137/0703001**[5]**K. O. Friedrichs and H. B. Keller,*A finite difference scheme for generalized Neumann problems*, Numerical Solution of Partial Differential Equations (Proc. Sympos. Univ. Maryland, 1965) Academic Press, New York, 1966, pp. 1–19. MR**0203956****[6]**Hiroshi Fujita,*On the nonlinear equations Δ𝑢+𝑒^{𝑢}=0 and ∂𝑣/∂𝑡=Δ𝑣+𝑒^{𝑣}*, Bull. Amer. Math. Soc.**75**(1969), 132–135. MR**0239258**, https://doi.org/10.1090/S0002-9904-1969-12175-0**[7]**I. M. Gel′fand,*Some problems in the theory of quasi-linear equations*, Uspehi Mat. Nauk**14**(1959), no. 2 (86), 87–158 (Russian). MR**0110868****[8]**Eugene Isaacson and Herbert Bishop Keller,*Analysis of numerical methods*, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR**0201039****[9]**D. D. Joseph, ``Nonlinear heat generation and stability of the temperature distribution in conduction solids,''*Int. J. Heat Mass Transfer*, v. 8, 1965, pp. 281-288.**[10]**D. D. Joseph and E. M. Sparrow,*Nonlinear diffusion induced by nonlinear sources*, Quart. Appl. Math.**28**(1970), 327–342. MR**0272272**, https://doi.org/10.1090/S0033-569X-1970-0272272-0**[11]**Herbert B. Keller,*Elliptic boundary value problems suggested by nonlinear diffusion processes*, Arch. Rational Mech. Anal.**35**(1969), 363–381. MR**0255979**, https://doi.org/10.1007/BF00247683**[12]**Herbert B. Keller,*Newton’s method under mild differentiability conditions*, J. Comput. System Sci.**4**(1970), 15–28. MR**0250476**, https://doi.org/10.1016/S0022-0000(70)80009-5**[13]**Herbert B. Keller and Donald S. Cohen,*Some positone problems suggested by nonlinear heat generation*, J. Math. Mech.**16**(1967), 1361–1376. MR**0213694****[14]**James R. Kuttler,*Finite difference approximations for eigenvalues of uniformly elliptic operators*, SIAM J. Numer. Anal.**7**(1970), 206–232. MR**0273841**, https://doi.org/10.1137/0707014**[15]**Theodore Laetsch,*A note on a paper of Keller and Cohen*, J. Math. Mech.**18**(1968/1969), 1095–1100. MR**0248464****[16]**Ja. G. Panovko and I. I. Gubanova,*\cyr Ustoĭchivost′ i kolebaniya uprugikh sistem*, “Nauka”, Moscow, 1979 (Russian). \cyr Sovremennye kontseptsii, paradoksy i oshibki. [New concepts, paradoxes and fallacies]; Third edition, revised. MR**566209****[17]**R. B. Simpson,*Finite difference methods for mildly nonlinear eigenvalue problems.*, SIAM J. Numer. Anal.**8**(1971), 190–211. MR**0286269**, https://doi.org/10.1137/0708021**[18]**R. Bruce Simpson and Donald S. Cohen,*Positive solutions of nonlinear elliptic eigenvalue problems*, J. Math. Mech.**19**(1969/70), 895–910. MR**0435566****[19]**J. B. Rosen,*Approximate solution and error bounds for quasi-linear elliptic boundary value problems*, SIAM J. Numer. Anal.**7**(1970), 80–103. MR**0264861**, https://doi.org/10.1137/0707004**[20]**L. V. Kantorovič and G. P. Akilov,*\cyr Funktsional′nyĭ analiz v normirovannykh prostranstvakh*, Gosudarstv. Izdat. Fis.-Mat. Lit., Moscow, 1959 (Russian). MR**0119071**

Retrieve articles in *Mathematics of Computation*
with MSC:
65N25

Retrieve articles in all journals with MSC: 65N25

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1972-0315918-9

Keywords:
Finite-difference methods,
nonlinear eigenvalue problems,
nonlinear elliptic problems,
Laplacian,
bifurcation,
error estimates,
Newton method

Article copyright:
© Copyright 1972
American Mathematical Society