Semidiscrete leastsquares methods for second order parabolic problems with nonhomogenous data
Author:
J. Thomas King
Journal:
Math. Comp. 28 (1974), 405411
MSC:
Primary 65N30
MathSciNet review:
0373323
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: Recently, Bramble and Thomée proposed semidiscrete leastsquares methods for the heat equation. In this paper we extend these methods to variable coefficient parabolic operators with nonhomogeneous equations and boundary conditions.
 [1]
S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Math. Studies, no. 2, Van Nostrand, Princeton, N. J., 1965. MR 31 #2504. MR 0178246 (31:2504)
 [2]
I. Babuška, Approximation by Hill Functions, University of Maryland Technical Note BN648, 1970.
 [3]
J. H. Bramble & A. H. Schatz, "RayleighRitzGalerkin methods for Dirichlet's problem using subspaces without boundary conditions," Comm. Pure Appl. Math., v. 23, 1970, pp. 653675. MR 42 #2690. MR 0267788 (42:2690)
 [4]
J. H. Bramble & S. R. Hilbert, "Bounds for a class of linear functionals with applications to Hermite interpolation," Numer. Math., v. 16, 1970, pp. 362369. MR 44 #7704. MR 0290524 (44:7704)
 [5]
J. H. Bramble & S. R. Hilbert, "Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation," SIAM J. Numer. Anal., v. 7, 1970, pp. 112124. MR 41 #7819. MR 0263214 (41:7819)
 [6]
J. H. Bramble & V. Thomée, "Semidiscrete leastsquares methods for a parabolic boundary value problem," Math. Comp., v. 26, 1972, pp. 633648. MR 0349038 (50:1532)
 [7]
J. H. Bramble & M. Zlamal, "Triangular elements in the finite element method," Math. Comp., v. 24, 1970, pp. 809820. MR 43 #8250. MR 0282540 (43:8250)
 [8]
J. Douglas & T. Dupont, "Galerkin methods for parabolic equations," SIAM J. Numer. Anal., v. 7, 1970, pp. 575626. MR 43 #2863. MR 0277126 (43:2863)
 [9]
S. Hilbert, Numerical Methods for Elliptic Boundary Value Problems, Doctoral Thesis, University of Maryland, College Park, Md., 1969.
 [10]
J. T. King, "The approximate solution of parabolic initialboundary value problems by weighted leastsquares methods," SIAM J. Numer. Anal., v. 9, 1972, pp. 215229. MR 0305626 (46:4756)
 [11]
O. A. Ladyženskaja, V. A. Solonnikov & N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, "Nauka", Moscow, 1967; English transl., Transl. Math. Monographs, vol. 23, Amer. Math. Soc., Providence, R.I., 1968. MR 39 #3159a, b. MR 0241822 (39:3159b)
 [12]
H. S. Price & R. S. Varga, Error Bounds for Semidiscrete Approximations of Parabolic Problems with Applications to Petroleum Reservoir Mechanics, SIAMAMS Proc., vol. II, Amer. Math. Soc., Providence, R.I., 1970, pp. 7494. MR 42 #1358. MR 0266452 (42:1358)
 [13]
M. H. Schultz, "Approximation theory of multivariate spline functions in Sobolev spaces," SIAM J. Numer. Anal., v. 6, 1969, pp. 570582. MR 41 #7823. MR 0263218 (41:7823)
 [14]
B. Swartz & B. Wendroff, "Generalized finite difference schemes," Math. Comp., v. 23, 1969, pp. 3749. MR 39 #1125. MR 0239768 (39:1125)
 [1]
 S. Agmon, Lectures on Elliptic Boundary Value Problems, Van Nostrand Math. Studies, no. 2, Van Nostrand, Princeton, N. J., 1965. MR 31 #2504. MR 0178246 (31:2504)
 [2]
 I. Babuška, Approximation by Hill Functions, University of Maryland Technical Note BN648, 1970.
 [3]
 J. H. Bramble & A. H. Schatz, "RayleighRitzGalerkin methods for Dirichlet's problem using subspaces without boundary conditions," Comm. Pure Appl. Math., v. 23, 1970, pp. 653675. MR 42 #2690. MR 0267788 (42:2690)
 [4]
 J. H. Bramble & S. R. Hilbert, "Bounds for a class of linear functionals with applications to Hermite interpolation," Numer. Math., v. 16, 1970, pp. 362369. MR 44 #7704. MR 0290524 (44:7704)
 [5]
 J. H. Bramble & S. R. Hilbert, "Estimation of linear functionals on Sobolev spaces with applications to Fourier transforms and spline interpolation," SIAM J. Numer. Anal., v. 7, 1970, pp. 112124. MR 41 #7819. MR 0263214 (41:7819)
 [6]
 J. H. Bramble & V. Thomée, "Semidiscrete leastsquares methods for a parabolic boundary value problem," Math. Comp., v. 26, 1972, pp. 633648. MR 0349038 (50:1532)
 [7]
 J. H. Bramble & M. Zlamal, "Triangular elements in the finite element method," Math. Comp., v. 24, 1970, pp. 809820. MR 43 #8250. MR 0282540 (43:8250)
 [8]
 J. Douglas & T. Dupont, "Galerkin methods for parabolic equations," SIAM J. Numer. Anal., v. 7, 1970, pp. 575626. MR 43 #2863. MR 0277126 (43:2863)
 [9]
 S. Hilbert, Numerical Methods for Elliptic Boundary Value Problems, Doctoral Thesis, University of Maryland, College Park, Md., 1969.
 [10]
 J. T. King, "The approximate solution of parabolic initialboundary value problems by weighted leastsquares methods," SIAM J. Numer. Anal., v. 9, 1972, pp. 215229. MR 0305626 (46:4756)
 [11]
 O. A. Ladyženskaja, V. A. Solonnikov & N. N. Ural'ceva, Linear and Quasilinear Equations of Parabolic Type, "Nauka", Moscow, 1967; English transl., Transl. Math. Monographs, vol. 23, Amer. Math. Soc., Providence, R.I., 1968. MR 39 #3159a, b. MR 0241822 (39:3159b)
 [12]
 H. S. Price & R. S. Varga, Error Bounds for Semidiscrete Approximations of Parabolic Problems with Applications to Petroleum Reservoir Mechanics, SIAMAMS Proc., vol. II, Amer. Math. Soc., Providence, R.I., 1970, pp. 7494. MR 42 #1358. MR 0266452 (42:1358)
 [13]
 M. H. Schultz, "Approximation theory of multivariate spline functions in Sobolev spaces," SIAM J. Numer. Anal., v. 6, 1969, pp. 570582. MR 41 #7823. MR 0263218 (41:7823)
 [14]
 B. Swartz & B. Wendroff, "Generalized finite difference schemes," Math. Comp., v. 23, 1969, pp. 3749. MR 39 #1125. MR 0239768 (39:1125)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65N30
Retrieve articles in all journals
with MSC:
65N30
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718197403733235
PII:
S 00255718(1974)03733235
Keywords:
Semidiscrete leastsquares method,
error analysis,
parabolic equation
Article copyright:
© Copyright 1974
American Mathematical Society
