Simplified proofs of error estimates for the least squares method for Dirichlet's problem

Author:
Garth A. Baker

Journal:
Math. Comp. **27** (1973), 229-235

MSC:
Primary 65N15

DOI:
https://doi.org/10.1090/S0025-5718-1973-0327056-0

MathSciNet review:
0327056

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Abstract: Recently, Bramble and Schatz have proposed a projection method for approximating the solution of Dirichlet's problem. Error estimates are derived by the authors using arguments based on certain interpolation theorems for linear operators on Hilbert spaces.

It is shown here that simpler and shorter methods can be used to obtain these error estimates.

**[1]**J. H. Bramble & A. H. Schatz, "Rayleigh-Ritz-Galerkin methods for Dirichlet's problem using subspaces without boundary conditions,"*Comm. Pure Appl. Math.*, v. 23, 1970, pp. 653-675. MR**42**#2690. MR**0267788 (42:2690)****[2]**J. H. Bramble, T. Dupont & V. Thomée,*Higher Order Polygonal Domain Galerkin Approximations in Dirichlet's Problem*, MRC Technical Summary Report #1213, March 1972.**[3]**J. H. Bramble & S. R. 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, "Semidiscrete-least squares methods for a parabolic boundary value problem,"*Math. Comp.*, v. 26, 1972, pp. 633-648. MR**0349038 (50:1532)****[5]**J. L. Lions & E. Magènes,*Problèmes aux Limites non Homogènes et Applications*. Vol. 1, Travaux et Recherches Mathématiques, no. 17, Dunod, Paris, 1968. MR**40**#512. MR**0247243 (40:512)****[5]**M. Schechter, "On estimates and regularity. II,"*Math. Scand.*, v. 13, 1963, pp. 47-69. MR**32**#6052. MR**0188616 (32:6052)****[7]**V. Thomée, Private communication.

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DOI:
https://doi.org/10.1090/S0025-5718-1973-0327056-0

Article copyright:
© Copyright 1973
American Mathematical Society