Semidiscrete and single step fully discrete approximations for second order hyperbolic equations with timedependent coefficients
Author:
Laurence A. Bales
Journal:
Math. Comp. 43 (1984), 383414
MSC:
Primary 65M60; Secondary 65M20
MathSciNet review:
758190
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: norm error estimates are proved for finite element approximations to the solutions of initial boundary value problems for second order hyperbolic partial differential equations with timedependent coefficients. Optimal order rates of convergence are shown for semidiscrete and single step fully discrete schemes using specially constructed initial data. The initial data are designed so that the data used for the fully discrete equation is reasonable to compute and so that the optimal order estimates can be proved.
 [1]
Garth
A. Baker, Error estimates for finite element methods for second
order hyperbolic equations, SIAM J. Numer. Anal. 13
(1976), no. 4, 564–576. MR 0423836
(54 #11810)
 [2]
Garth
A. Baker and James
H. Bramble, Semidiscrete and single step fully discrete
approximations for second order hyperbolic equations, RAIRO Anal.
Numér. 13 (1979), no. 2, 75–100
(English, with French summary). MR 533876
(80f:65115)
 [3]
Garth
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/S0025571819770448947X
 [4]
Garth
A. Baker, Vassilios
A. Dougalis, and Steven
M. Serbin, High order accurate twostep approximations for
hyperbolic equations, RAIRO Anal. Numér. 13
(1979), no. 3, 201–226 (English, with French summary). MR 543933
(81c:65044)
 [5]
James
H. Bramble and Peter
H. Sammon, Efficient higher order single step
methods for parabolic problems. I, Math.
Comp. 35 (1980), no. 151, 655–677. MR 572848
(81h:65110), http://dx.doi.org/10.1090/S0025571819800572848X
 [6]
J.
H. Bramble, A.
H. Schatz, V.
Thomée, and L.
B. Wahlbin, Some convergence estimates for semidiscrete Galerkin
type approximations for parabolic equations, SIAM J. Numer. Anal.
14 (1977), no. 2, 218–241. MR 0448926
(56 #7231)
 [7]
M. Crouzeix, Sur l'Approximation des Équations Différentielles Opérationnelles Linéaires par des Méthodes de RungeKutta, Thèse, Université de Paris VI, 1975.
 [8]
Vassilios
A. Dougalis, MultistepGalerkin methods for
hyperbolic equations, Math. Comp.
33 (1979), no. 146, 563–584. MR 521277
(81b:65081), http://dx.doi.org/10.1090/S00255718197905212775
 [9]
Vassilios
A. Dougalis and Steven
M. Serbin, Twostep highorder accurate full discretizations of
secondorder hyperbolic equations, Advances in computer methods for
partial differential equations, III (Proc. Third IMACS Internat. Sympos.,
Lehigh Univ., Bethlehem, Pa., 1979), IMACS, New Brunswick, N.J., 1979,
pp. 214–220. MR 603474
(82b:65080)
 [10]
Jim
Douglas Jr., Todd
Dupont, and Richard
E. Ewing, Incomplete iteration for timestepping a Galerkin method
for a quasilinear parabolic problem, SIAM J. Numer. Anal.
16 (1979), no. 3, 503–522. MR 530483
(80f:65117), http://dx.doi.org/10.1137/0716039
 [11]
Todd
Dupont, 𝐿²estimates for Galerkin methods for second
order hyperbolic equations, SIAM J. Numer. Anal. 10
(1973), 880–889. MR 0349045
(50 #1539)
 [12]
E.
Gekeler, Linear multistep methods and Galerkin procedures for
initial boundary value problems, SIAM J. Numer. Anal.
13 (1976), no. 4, 536–548. MR 0431749
(55 #4744)
 [13]
E.
Gekeler, GalerkinRungeKutta methods and hyperbolic initial
boundary value problems, Computing 18 (1977),
no. 1, 79–88 (English, with German summary). MR 0438739
(55 #11646)
 [14]
Gianni
Gilardi, Teoremi di regolarità per la soluzione di
un’equazione differenziale astratta lineare del secondo ordine,
Ist. Lombardo Accad. Sci. Lett. Rend. A 106 (1972),
641–675 (Italian). MR 0333386
(48 #11711)
 [15]
Louis
A. Hageman and David
M. Young, Applied iterative methods, Academic Press, Inc.
[Harcourt Brace Jovanovich, Publishers], New YorkLondon, 1981. Computer
Science and Applied Mathematics. MR 630192
(83c:65064)
 [16]
J.
D. Lambert, Computational methods in ordinary differential
equations, John Wiley & Sons, LondonNew YorkSydney, 1973.
Introductory Mathematics for Scientists and Engineers. MR 0423815
(54 #11789)
 [17]
J. L. Lions, E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. I, SpringerVerlag, Berlin and New York, 1972.
 [18]
J. L. Lions & E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. II, SpringerVerlag, Berlin and New York, 1972.
 [19]
Ming
You Huang and Vidar
Thomée, Some convergence estimates for
semidiscrete type schemes for timedependent nonselfadjoint parabolic
equations, Math. Comp. 37
(1981), no. 156, 327–346. MR 628699
(82i:65060), http://dx.doi.org/10.1090/S00255718198106286991
 [20]
P. H. Sammon, Approximations for Parabolic Equations with TimeDependent Coefficients, Ph.D. Thesis, Cornell University, 1978.
 [21]
Peter
H. Sammon, Convergence estimates for semidiscrete parabolic
equation approximations, SIAM J. Numer. Anal. 19
(1982), no. 1, 68–92. MR 646595
(83g:65094), http://dx.doi.org/10.1137/0719002
 [22]
Miloš
Zlámal, Finite element multistep
discretizations of parabolic boundary value problems, Math. Comp. 29 (1975), 350–359. MR 0371105
(51 #7326), http://dx.doi.org/10.1090/S00255718197503711052
 [1]
 G. A. Baker, "Error estimates for finite element methods for second order hyperbolic equations," SIAM J. Numer. Anal., v. 13, 1976, pp. 564576. MR 0423836 (54:11810)
 [2]
 G. A. Baker & J. H. Bramble, "Semidiscrete and single step fully discrete approximations for second order hyperbolic equations," RAIRO Anal. Numér., v. 13, 1979, pp. 75100. MR 533876 (80f:65115)
 [3]
 G. A. Baker, J. H. Bramble & V. Thomée, "Single step Galerkin approximations for parabolic problems," Math. Comp., v. 31, 1977, pp. 818847. MR 0448947 (56:7252)
 [4]
 G. A. Baker, V. A. Dougalis & S. M. Serbin, "High order accurate twostep approximations for hyperbolic equations," RAIRO Anal. Numér., v. 13, 1979, pp. 201206. MR 543933 (81c:65044)
 [5]
 J. H. Bramble & P. H. Sammon, "Efficient higher order single step methods for parabolic problems: Part I," Math. Comp., v. 35, 1980, pp. 655677. MR 572848 (81h:65110)
 [6]
 J. H. Bramble, A. H. Schatz, V. Thomée & L. B. Wahlbin, "Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations," SIAM J. Numer. Anal., v. 14, 1977, pp. 218241. MR 0448926 (56:7231)
 [7]
 M. Crouzeix, Sur l'Approximation des Équations Différentielles Opérationnelles Linéaires par des Méthodes de RungeKutta, Thèse, Université de Paris VI, 1975.
 [8]
 V. A. Dougalis, "Multistep Galerkin methods for hyperbolic equations," Math. Comp., v. 33, 1979, pp. 563584. MR 521277 (81b:65081)
 [9]
 V. A. Dougalis & S. M. Serbin, "Twostep high order accurate full discretizations of second order hyperbolic equations," Proc. 3rd IMACS Symposium, Advances in Computer Methods for Partial Differential Equations (R. Vichnevetsky and R. S. Stepleman, eds.), IMACS, 1979, pp. 214220. MR 603474 (82b:65080)
 [10]
 J. Douglas, Jr., T. Dupont & R. E. Ewing, "Incomplete iteration for timestepping a Galerkin method for a quasilinear parabolic problem," SIAM J. Numer. Anal., v. 16, 1979, pp. 503522. MR 530483 (80f:65117)
 [11]
 T. Dupont, "estimates for Galerkin methods for second order hyperbolic equations," SIAM J. Numer. Anal., v. 10, 1973, pp. 880889. MR 0349045 (50:1539)
 [12]
 E. Gekeler, "Linear multistep methods and Galerkin procedures for initial boundary value problems," SIAM J. Numer. Anal., v. 13, 1976, pp. 536548. MR 0431749 (55:4744)
 [13]
 E. Gekeler, "GalerkinRungeKutta methods and hyperbolic initial boundary value problems," Computing, v. 18, 1977, pp. 7988. MR 0438739 (55:11646)
 [14]
 G. Gilardi, "Teoremi di regolarità per la soluzione di un'equazione differenziale astratta lineare del secondo ordine," Istit. Lombardo Accad. Sci. Lett. Rend. A, v. 106, 1972, pp. 641675. MR 0333386 (48:11711)
 [15]
 L. A. Hageman & D. M. Young, Applied Iterative Methods, Academic Press, New York, 1981. MR 630192 (83c:65064)
 [16]
 J. D. Lambert, Computational Methods in Ordinary Differential Equations, Wiley, New York, 1973. MR 0423815 (54:11789)
 [17]
 J. L. Lions, E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. I, SpringerVerlag, Berlin and New York, 1972.
 [18]
 J. L. Lions & E. Magenes, Nonhomogeneous Boundary Value Problems and Applications, Vol. II, SpringerVerlag, Berlin and New York, 1972.
 [19]
 H. Mingyou & V. Thomée, "Some convergence estimates for semidiscrete type schemes for timedependent nonselfadjoint parabolic equations," Math. Comp., v. 37, 1981, pp. 327346. MR 628699 (82i:65060)
 [20]
 P. H. Sammon, Approximations for Parabolic Equations with TimeDependent Coefficients, Ph.D. Thesis, Cornell University, 1978.
 [21]
 P. H. Sammon, "Convergence estimates for semidiscrete parabolic equation approximations," SIAM J. Numer. Anal., v. 19, 1982, pp. 6892. MR 646595 (83g:65094)
 [22]
 M. Zlámal, "Finite element multistep discretizations of parabolic boundary value problems," Math. Comp., v. 29, 1975, pp. 350359. MR 0371105 (51:7326)
Similar Articles
Retrieve articles in Mathematics of Computation
with MSC:
65M60,
65M20
Retrieve articles in all journals
with MSC:
65M60,
65M20
Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718198407581906
PII:
S 00255718(1984)07581906
Article copyright:
© Copyright 1984
American Mathematical Society
