Logarithmic convexity for discrete harmonic functions and the approximation of the Cauchy problem for Poisson's equation

Authors:
R. S. Falk and P. B. Monk

Journal:
Math. Comp. **47** (1986), 135-149

MSC:
Primary 65M10; Secondary 35R35, 65M30

DOI:
https://doi.org/10.1090/S0025-5718-1986-0842126-5

MathSciNet review:
842126

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Abstract | References | Similar Articles | Additional Information

Abstract: Logarithmic convexity type continuous dependence results for discrete harmonic functions defined as solutions of the standard piecewise-linear approximation to Laplace's equation are proved. Using this result, error estimates for a regularization method for approximating the Cauchy problem for Poisson's equation on a rectangle are obtained. Numerical results are presented.

**[1]**J. R. Cannon, "Error estimates for some unstable continuation problems,"*J. Soc. Indust. Appl. Math.*, v. 12, 1964, pp. 270-284. MR**0168897 (29:6153)****[2]**J. R. Cannon & K. Miller, "Some problems in numerical analytic continuation,"*SIAM J. Numer. Anal.*, v. 2, 1965, pp. 87-98. MR**0179908 (31:4145)****[3]**J. R. Cannon & J. Douglas, Jr., "The approximation of harmonic and parabolic functions on half-spaces from interior data,"*Numerical Analysis of Partial Differential Equations*(C.I.M.E. 2 Ciclo, Ispra, 1967), Edizioni Cremonese, Rome, 1968, pp. 193-230. MR**0243755 (39:5076)****[4]**J. Douglas, Jr., "A numerical method for analytic continuation,"*Boundary Value Problems in Differential Equations*, University of Wisconsin Press, 1960, pp. 179-189. MR**0117866 (22:8640)****[5]**P. Franzone & E. Magenes, "On the inverse potential problem of electrocardiology,"*Calcolo*, v. 16, 1979, pp. 459-538. MR**592483 (82c:35079)****[6]**P. Grisvard,*Elliptic Problems in Non-Smooth Domains*, Pitman, New York, 1985.**[7]**Houde Han, "The finite element method in a family of improperly posed problems,"*Math. Comp.*, v. 38, 1982, pp. 55-65. MR**637286 (82m:65091)****[8]**L. E. Payne, "Bounds in the Cauchy problem for Laplace's equation,"*Arch. Rational Mech. Anal.*, v. 5, 1960, pp. 35-45. MR**0110875 (22:1743)****[9]**L. E. Payne, "On a priori bounds in the Cauchy problem for elliptic equations,"*SIAM J. Math. Anal.*, v. 1, 1970, pp. 82-89. MR**0262654 (41:7259)****[10]**L. E. Payne,*Improperly Posed Problems in Partial Differential Equations*, SIAM, Philadelphia, Pa., 1975. MR**0463736 (57:3678)****[11]**R. Rannacher & R. Scott, "Some optimal error estimates for piecewise linear finite element approximations,"*Math. Comp.*, v, 38, 1982, pp. 437-445. MR**645661 (83e:65180)****[12]**R. Scott, "Interpolated boundary conditions in the finite element method,"*SIAM. J. Numer. Anal.*, v. 12, 1975, pp. 404-427. MR**0386304 (52:7162)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1986-0842126-5

Keywords:
Ill-posed problems,
logarithmic convexity,
Poisson's equation

Article copyright:
© Copyright 1986
American Mathematical Society