Efficient higher order single step methods for parabolic problems. I
James H. Bramble and Peter H. Sammon
Math. Comp. 35 (1980), 655-677
Primary 65N30; Secondary 65M15
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Abstract: Some efficient, high order methods are discussed for approximating the solution of an initial boundary value problem for a homogeneous parabolic equation with time dependent coefficients. The methods are based on Galerkin-type approximations in the spacial variables and single step methods in the time variable. The equations defining the time-stepping procedure are solved only approximately, however. A preconditioned iterative technique is used for this purpose. The resulting algorithm is shown to produce optimal order approximations using only the order of work required by the single step method applied to the parabolic problem with time independent coefficients.
O. AXELSSON, On Preconditioning and Convergence Acceleration in Sparse Matrix Problems, CERN (European Organization for Nuclear Research), Geneva, 1974.
A. Baker, James
H. Bramble, and Vidar
Thomée, Single step Galerkin approximations
for parabolic problems, Math. Comp.
31 (1977), no. 140, 818–847. MR 0448947
(56 #7252), http://dx.doi.org/10.1090/S0025-5718-1977-0448947-X
Douglas Jr. and Todd
Dupont, Alternating-direction Galerkin methods on rectangles,
Numerical Solution of Partial Differential Equations, II (SYNSPADE 1970)
(Proc. Sympos., Univ. of Maryland, College Park, Md., 1970) Academic
Press, New York, 1971, pp. 133–214. MR 0273830
Douglas Jr., Todd
Dupont, and Richard
E. Ewing, Incomplete iteration for time-stepping a Galerkin method
for a quasilinear parabolic problem, SIAM J. Numer. Anal.
16 (1979), no. 3, 503–522. MR 530483
Friedman, Partial differential equations, Corrected reprint of
the original edition, Robert E. Krieger Publishing Co., Huntington, N.Y.,
0454266 (56 #12517)
Lions and E.
Magenes, Non-homogeneous boundary value problems and applications.
Vol. III, Springer-Verlag, New York-Heidelberg, 1973. Translated from
the French by P. Kenneth; Die Grundlehren der mathematischen
Wissenschaften, Band 183. MR 0350179
R. Nassif and Jean
Descloux, Stability study for time-dependent linear parabolic
equations and its application to Hermitian methods, Topics in
numerical analysis, III (Proc. Roy. Irish Acad. Conf., Trinity Coll.,
Dublin, 1976) Academic Press, London, 1977, pp. 293–316. MR 0657787
P. H. SAMMON, Convergence Estimates for Semidiscrete Parabolic Equation Approximations, Mathematics Research Center Technical Survey Report No. 2053, 1980.
S. Varga, Matrix iterative analysis, Prentice-Hall, Inc.,
Englewood Cliffs, N.J., 1962. MR 0158502
- O. AXELSSON, On Preconditioning and Convergence Acceleration in Sparse Matrix Problems, CERN (European Organization for Nuclear Research), Geneva, 1974.
- G. A. BAKER, J. H. BRAMBLE & V. THOMÉE, "Single step Galerkin approximations for parabolic problems," Math. Comp., v. 31. 1977, pp. 818-847. MR 0448947 (56:7252)
- J. DOUGLAS, JR. & T. DUPONT, "Alternating direction methods on rectangles," Numerical Solution of Partial Differential Equations-II (B. Hubbard, Ed.), Academic Press, New York, 1971. MR 0273830 (42:8706)
- J. DOUGLAS, JR., T. DUPONT & R. EWING, "Incomplete iteration for time-stepping a Galerkin method for a quasilinear parabolic problem," SIAM J. Numer. Anal., v. 16, 1979, pp. 503-522. MR 530483 (80f:65117)
- A. FRIEDMAN, Partial Differential Equations, Krieger, Huntington, New York, 1976. MR 0454266 (56:12517)
- J. L. LIONS & E. MAGENES, Nonhomogeneous Boundary Value Problems and Applications, Vol. II, Springer-Verlag, New York, 1973. MR 0350179 (50:2672)
- N. NASSIF & J. DESCLOUX, "Stability study for time-dependent linear parabolic equations and its application to Hermitian methods," Topics in Numerical Analysis III (J. Miller, Ed.), Academic Press, New York, 1977. MR 0657787 (58:31875)
- P. H. SAMMON, Convergence Estimates for Semidiscrete Parabolic Equation Approximations, Mathematics Research Center Technical Survey Report No. 2053, 1980.
- RICHARD S. VARGA, Matrix Iterative Analysis, Prentice-Hall, Englewood Cliffs, N.J., 1962. MR 0158502 (28:1725)
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