Finite difference discretizations of some initial and boundary value problems with interface
Authors:
Georgios D. Akrivis and Vassilios A. Dougalis
Journal:
Math. Comp. 56 (1991), 505522
MSC:
Primary 65M12; Secondary 65M15
MathSciNet review:
1066829
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Abstract: We analyze the discretization of initial and boundary value problems with a stationary interface in one space dimension for the heat equation, the Schrödinger equation, and the wave equation by finite difference methods. Extending the concept of the elliptic projection, well known from the analysis of Galerkin finite element methods, to our finite difference case, we prove secondorder error estimates in space and time in the norm.
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Eugene
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, Homogeneous difference schemes on nonuniform nets for equations of parabolic type, U.S.S.R. Comput. Math. and Math. Phys. 3 (1963), 351393.
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A. Samarskii and I.
V. Fryazinov, On the convergence of difference schemes for a
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Mary
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 [1]
 G. D. Akrivis and V. A. Dougalis, On a class of conservative, highly accurate Galerkin methods for the Schrödinger equation, RAIRO Modél. Math. Anal. Numér. (to appear). MR 1135988 (92m:65117)
 [2]
 L. A. Bales, V. A. Dougalis, and S. M. Serbin, Cosine methods for secondorder hyperbolic equations with timedependent coefficients, Math. Comp. 45 (1985), 6589. MR 790645 (86j:65112)
 [3]
 B. M. Budak, Homogeneous differentialdifference schemes of second order accuracy for parabolic and hyperbolic equations with discontinuous coefficients, Soviet Math. Dokl. 3 (1962), 198202. MR 0144483 (26:2027)
 [4]
 E. C. Gartland, Jr., Compact highorder finite differences for interface problems in one dimension, IMA J. Numer. Anal. 9 (1989), 243260. MR 1000460 (91a:65193)
 [5]
 E. Isaacson, Error estimates for parabolic equations, Comm. Pure Appl. Math. 24 (1961), 381389. MR 0137311 (25:763)
 [6]
 S. T. McDaniel, Applications of energy methods to finitedifference solutions of the parabolic wave equation, Comput. Math. Appl. 11 (1985), 823829. MR 809614 (86k:65079)
 [7]
 S. T. McDaniel and D. Lee, A finite difference treatment of interface conditions for the parabolic wave equation: the horizontal interface, J. Acoust. Soc. Amer. 71 (1982), 855858. MR 651838 (83d:65323)
 [8]
 A. A. Samarskii, A priori estimates for difference equations, U.S.S.R. Comput. Math. and Math. Phys. 1 (1961), 11381167.
 [9]
 , Homogeneous difference schemes on nonuniform nets for equations of parabolic type, U.S.S.R. Comput. Math. and Math. Phys. 3 (1963), 351393.
 [10]
 A. A. Samarskii and I. V. Fryazinov, On the convergence of difference schemes for a heatconduction equation with discontinuous coefficients, U.S.S.R. Comput. Math. and Math. Phys. 1 (1961), 962982. MR 0205493 (34:5320)
 [11]
 A. A. Samarskij, Theorie der Differenzenverfahren, Akad. Verlag., Leipzig, 1984. MR 783639 (86e:65003)
 [12]
 V. Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Math., vol. 1054, SpringerVerlag, Berlin, Heidelberg, New York, Tokyo, 1984. MR 744045 (86k:65006)
 [13]
 R. S. Varga, Matrix iterative analysis, PrenticeHall, Englewood Cliffs, N. J., 1962. MR 0158502 (28:1725)
 [14]
 M. F. Wheeler, A priori error estimates for Galerkin approximations to parabolic partial differential equations, SIAM J. Numer. Anal. 10 (1973), 723759. MR 0351124 (50:3613)
 [15]
 , estimates of optimal orders for Galerkin methods for onedimensional second order parabolic and hyperbolic equations, SIAM J. Numer. Anal. 10 (1973), 908913. MR 0343658 (49:8398)
 [16]
 , An optimal error estimate for Galerkin approximations to solutions of twopoint boundary value problems, SIAM J. Numer. Anal. 10 (1973), 914917. MR 0343659 (49:8399)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199110668297
PII:
S 00255718(1991)10668297
Article copyright:
© Copyright 1991
American Mathematical Society
