Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)



A least squares procedure for the wave equation

Author: Alfred Carasso
Journal: Math. Comp. 28 (1974), 757-767
MSC: Primary 65N05
MathSciNet review: 0373310
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We develop and analyze a least squares procedure for approximating the homogeneous Dirichlet problem for the wave equation in a bounded domain $ \Omega $ in $ {R^N}$. This procedure is based on the pure implicit scheme for time differencing. Surprisingly, it is the normal derivative of u rather than u itself which must be included in the boundary functional. This normal derivative is an unknown quantity. We show that it may be set equal to zero while retaining the $ O(k)$ accuracy of the pure implicit scheme. The penalty is that one must use smoother trial functions to obtain this accuracy.

References [Enhancements On Off] (What's this?)

  • [1] S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Math. Studies, no. 2, Van Nostrand, New York, 1965. MR 31 #2504. MR 0178246 (31:2504)
  • [2] G. Birkhoff, M. Schultz & R. Varga, "Piecewise Hermite interpolation in one and two variables with applications to partial differential equations," Numer. Math., v. 11, 1968, pp. 232-256. MR 37 #2404, MR 0226817 (37:2404)
  • [3] J. H. Bramble & S. Hilbert, "Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation," SIAM J. Numer. Anal., v. 7, 1970, pp. 112-124. MR 41 #7819. MR 0263214 (41:7819)
  • [4] J. H. Bramble & V. Thomée, "Semi-discrete least squares methods for a parabolic boundary value problem," Math. Comp. v. 26, 1972, pp. 633-648. MR 0349038 (50:1532)
  • [5] T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der math. Wissenschaften, Band 132, Springer-Verlag, New York, 1966. MR 34 #3324. MR 0203473 (34:3324)
  • [6] J. L. Lions & E. Magenes, Problèmes aux limites homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, no. 17, Dunod, Paris, 1968. MR 40 #512. MR 0247243 (40:512)
  • [7] S. M. Serbin, Doctoral Dissertation, Department of Mathematics, Cornell University, Ithaca, N.Y., 1971.
  • [8] R. S. Varga, Functional Analysis and Approximation Theory in Numerical Analysis, Regional Conference Series in Appl. Math., vol. 3, SIAM, Philadelphia, Pa., 1971. MR 0310504 (46:9602)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 65N05

Retrieve articles in all journals with MSC: 65N05

Additional Information

Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society