The finite element method in the family of improperly posed problems
Author: Hou De Han
Journal: Math. Comp. 38 (1982), 55-65
MSC: Primary 65M60; Secondary 35R25, 65M30
MathSciNet review: 637286
Full-text PDF Free Access
Abstract: The numerical solution of the Cauchy problem for elliptic equations is considered. We reformulate the original problem as a variational inequality problem, which we solve using the finite element method. Moreover, we prove the convergence of the approximate solution.
-  J. Hadamard, Lectures on the Cauchy Problem in Linear Partial Differential Equations, Yale Univ. Press, New Haven, 1923.
-  M. M. Lavrent′ev, On the Cauchy problem for Laplace equation, Izv. Akad. Nauk SSSR. Ser. Mat. 20 (1956), 819–842 (Russian). MR 0087874
-  L. E. Payne, Bounds in the Cauchy problem for the Laplace equation, Arch. Rational Mech. Anal. 5 (1960), 35–45 (1960). MR 0110875, https://doi.org/10.1007/BF00252897
-  L. E. Payne, On a priori bounds in the Cauchy problem for elliptic equations, SIAM J. Math. Anal. 1 (1970), 82–89. MR 0262654, https://doi.org/10.1137/0501008
-  L. E. Payne, Improperly posed problems in partial differential equations, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. Regional Conference Series in Applied Mathematics, No. 22. MR 0463736
Ekeland and Roger
Temam, Analyse convexe et problèmes variationnels,
Dunod; Gauthier-Villars, Paris-Brussels-Montreal, Que., 1974 (French).
Collection Études Mathématiques. MR
Ivar Ekeland and Roger Temam, Convex analysis and variational problems, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1976. Translated from the French; Studies in Mathematics and its Applications, Vol. 1. MR 0463994
-  Hou De Han and Long An Ying, An iterative method with infinite elements, Math. Numer. Sinica 1 (1979), no. 1, 91–99 (Chinese, with English summary). MR 656882
- J. Hadamard, Lectures on the Cauchy Problem in Linear Partial Differential Equations, Yale Univ. Press, New Haven, 1923.
- M. M. Lavrentiev, ``On the Cauchy problem for the Laplace equation,'' Izv. Akad. Nauk SSSR Ser. Mat., v. 20, 1956, pp. 819-842. (Russian) MR 0087874 (19:426d)
- L. E. Payne, ``Bounds in the Cauchy problem for the Laplace equation,'' Arch. Rational Mech. Anal., v. 5, 1960, pp. 35-45. MR 0110875 (22:1743)
- 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)
- L. E. Payne, Improperly Posed Problems in Partial Differential Equations, Regional Conf. Series in Appl. Math., SIAM, 1975. MR 0463736 (57:3678)
- I. Ekeland & R. Teman, Convex Analysis and Variational Problems, American Elsevier, New York, 1976. MR 0463994 (57:3931b)
- Hou-de Han & Long-an Ying, ``An iterative method in the infinite element,'' Math. Numer. Sinica, v. 1, 1979, pp. 91-99. MR 656882 (83d:65285)