Semidiscrete least-squares methods for a parabolic boundary value problem

Bramble, James H.; Thomée, Vidar

doi:10.1090/S0025-5718-1972-0349038-4

Semidiscrete least-squares methods for a parabolic boundary value problem
HTML articles powered by AMS MathViewer

by James H. Bramble and Vidar Thomée PDF

Math. Comp. 26 (1972), 633-648 Request permission

Abstract:

In this paper some approximate methods for solving the initial-boundary value problem for the heat equation in a cylinder under homogeneous boundary conditions are analyzed. The methods consist in discretizing with respect to time and solving approximately the resulting elliptic problem for fixed time by least squares methods. The approximate solutions will belong to a finite-dimensional subspace of functions in space which will not be required to satisfy the homogeneous boundary conditions.

References

Jean-Pierre Aubin, Interpolation et approximation optimales et “spline functions”, J. Math. Anal. Appl. 24 (1968), 1–24 (French). MR 231096, DOI 10.1016/0022-247X(68)90046-2

Calcolo

Ivo Babuška, The finite element method for elliptic equations with discontinuous coefficients, Computing (Arch. Elektron. Rechnen) 5 (1970), 207–213. MR 277119, DOI 10.1007/bf02248021
Approximation by hill functions, Comment. Math. Univ. Carolinae 11 (1970), 787–811. MR 292309
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
James H. Bramble and Alfred H. Schatz, Rayleigh-Ritz-Galerkin methods for Dirichlet’s problem using subspaces without boundary conditions, Comm. Pure Appl. Math. 23 (1970), 653–675. MR 267788, DOI 10.1002/cpa.3160230408
James H. Bramble and Miloš Zlámal, Triangular elements in the finite element method, Math. Comp. 24 (1970), 809–820. MR 282540, DOI 10.1090/S0025-5718-1970-0282540-0
F. Di Guglielmo, Construction d’approximations des espaces de Sobolev sur des réseaux en simplexes, Calcolo 6 (1969), 279–331. MR 433113, DOI 10.1007/BF02576159
Jim Douglas Jr. and Todd Dupont, Galerkin methods for parabolic equations, SIAM J. Numer. Anal. 7 (1970), 575–626. MR 277126, DOI 10.1137/0707048
George Fix and Gilbert Strang, Fourier analysis of the finite element method in Ritz-Galerkin theory, Studies in Appl. Math. 48 (1969), 265–273. MR 258297, DOI 10.1002/sapm1969483265
J. Thomas King, The approximate solution of parabolic initial boundary value problems by weighted least-squares methods, SIAM J. Numer. Anal. 9 (1972), 215–229. MR 305626, DOI 10.1137/0709020
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
Jaak Peetre and Vidar Thomée, On the rate of convergence for discrete initial-value problems, Math. Scand. 21 (1967), 159–176 (1969). MR 255085, DOI 10.7146/math.scand.a-10856
Harvey S. Price and Richard S. Varga, Error bounds for semidiscrete Galerkin approximations of parabolic problems with applications to petroleum reservoir mechanics, Numerical Solution of Field Problems in Continuum Physics (Proc. Sympos. Appl. Math., Durham, N.C., 1968) Amer. Math. Soc., Providence, R.I., 1970, pp. 74–94. MR 0266452
Martin H. Schultz, Rayleigh-Ritz-Galerkin methods for multidimensional problems, SIAM J. Numer. Anal. 6 (1969), 523–538. MR 263254, DOI 10.1137/0706047
Gilbert Strang, The finite element method and approximation theory, Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970) (Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic Press, New York, 1971, pp. 547–583. MR 0287723
Miloš Zlámal, On the finite element method, Numer. Math. 12 (1968), 394–409. MR 243753, DOI 10.1007/BF02161362