A least squares procedure for the wave equation

Carasso, Alfred

doi:10.1090/S0025-5718-1974-0373310-7

A least squares procedure for the wave equation
HTML articles powered by AMS MathViewer

by Alfred Carasso PDF

Math. Comp. 28 (1974), 757-767 Request permission

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

Shmuel Agmon, Lectures on elliptic boundary value problems, Van Nostrand Mathematical Studies, No. 2, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. Prepared for publication by B. Frank Jones, Jr. with the assistance of George W. Batten, Jr. MR 0178246
G. Birkhoff, M. H. Schultz, and R. S. Varga, Piecewise Hermite interpolation in one and two variables with applications to partial differential equations, Numer. Math. 11 (1968), 232–256. MR 226817, DOI 10.1007/BF02161845
J. H. Bramble and S. R. Hilbert, Estimation of linear functionals on Sobolev spaces with application to Fourier transforms and spline interpolation, SIAM J. Numer. Anal. 7 (1970), 112–124. MR 263214, DOI 10.1137/0707006
James H. Bramble and Vidar Thomée, Semidiscrete least-squares methods for a parabolic boundary value problem, Math. Comp. 26 (1972), 633–648. MR 349038, DOI 10.1090/S0025-5718-1972-0349038-4
Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
J.-L. Lions and E. Magenes, Problèmes aux limites non homogènes et applications. Vol. 1, Travaux et Recherches Mathématiques, No. 17, Dunod, Paris, 1968 (French). MR 0247243
R. S. Varga, Functional analysis and approximation theory in numerical analysis, Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 3, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1971. MR 0310504