Smoothing effect and discretization in time to semilinear parabolic equations with nonsmooth data

Marián Slodička

Commentationes Mathematicae Universitatis Carolinae (1991)

  • Volume: 32, Issue: 4, page 703-713
  • ISSN: 0010-2628

Abstract

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The purpose of this paper is to derive the error estimates for discretization in time of a semilinear parabolic equation in a Banach space. The estimates are given in the norm of the space 𝕏 α for 0 < α < 1 when the initial condition is not regular.

How to cite

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Slodička, Marián. "Smoothing effect and discretization in time to semilinear parabolic equations with nonsmooth data." Commentationes Mathematicae Universitatis Carolinae 32.4 (1991): 703-713. <http://eudml.org/doc/247302>.

@article{Slodička1991,
abstract = {The purpose of this paper is to derive the error estimates for discretization in time of a semilinear parabolic equation in a Banach space. The estimates are given in the norm of the space $\mathbb \{X\}_\{\alpha \}$ for $0<\alpha <1$ when the initial condition is not regular.},
author = {Slodička, Marián},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {error estimates; parabolic equation; backward Euler method; nonsmooth initial data; parabolic equation; nonsmooth initial data; method of lines; discretization in time; error estimates; abstract semilinear initial value problem; Banach space; Rothe method; backward Euler method},
language = {eng},
number = {4},
pages = {703-713},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Smoothing effect and discretization in time to semilinear parabolic equations with nonsmooth data},
url = {http://eudml.org/doc/247302},
volume = {32},
year = {1991},
}

TY - JOUR
AU - Slodička, Marián
TI - Smoothing effect and discretization in time to semilinear parabolic equations with nonsmooth data
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 1991
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 32
IS - 4
SP - 703
EP - 713
AB - The purpose of this paper is to derive the error estimates for discretization in time of a semilinear parabolic equation in a Banach space. The estimates are given in the norm of the space $\mathbb {X}_{\alpha }$ for $0<\alpha <1$ when the initial condition is not regular.
LA - eng
KW - error estimates; parabolic equation; backward Euler method; nonsmooth initial data; parabolic equation; nonsmooth initial data; method of lines; discretization in time; error estimates; abstract semilinear initial value problem; Banach space; Rothe method; backward Euler method
UR - http://eudml.org/doc/247302
ER -

References

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  1. Helfrich H.-P., Error estimates for semidiscrete Galerkin type approximations to semilinear evolution equations with nonsmooth initial data, Numer. Math. 51 (1987), 559-569. (1987) Zbl0648.65078MR0910865
  2. Henry D., Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer Verlag, Berlin-Heidelberg-New York, 1981. Zbl0663.35001MR0610244
  3. Johnson C., Larsson S., Thomee V., Wahlbin L.B., Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data, Math. Comp. 49 (1987), 331-357. (1987) Zbl0634.65110MR0906175
  4. Le Roux M.-N., Semidiscretization in time for parabolic problems, Math. Comp. 33 (1979), 919-931. (1979) Zbl0417.65049MR0528047
  5. Le Roux M.-N., Thomee V., Numerical solution of semilinear integrodifferential equations of parabolic type with nonsmooth data, SIAM J. Numer. Anal. 26 (1989), 1291-1309. (1989) Zbl0701.65091MR1025089
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  7. Mingyou H., Thomee V., On the backward Euler method for parabolic equations with rough initial data, SIAM J. Numer. Anal. 19 (1982), 599-603. (1982) Zbl0482.65051MR0656473
  8. Sammon P.H., Convergence estimates for semidiscrete parabolic equation approximations, SIAM J. Numer. Anal. 19 (1982), 68-92. (1982) Zbl0497.65056MR0646595
  9. Slodička M., Error estimates for discretization in time to linear homogeneous parabolic equations with nonsmooth initial data, preprint JINR, E5-90-08, Dubna, 1990. 
  10. Slodička M., Error estimates for discretization in time to nonhomogeneous parabolic equations with rough initial data, preprint JINR, E5-90-212, Dubna, 1990. 
  11. Taylor A.E., Introduction to Functional Analysis, Wiley, New York, 1958. Zbl0654.46002MR0098966
  12. Thomee V., Galerkin Finite Element Methods for Parabolic Problems, Lecture Notes in Math. 1054, Springer Verlag, Berlin-Heidelberg-New York-Tokyo, 1984. Zbl1105.65102MR0744045
  13. Thomee V., On the numerical solution of integro-differential equations of parabolic type, Inter. Ser. of Numer. Math., vol. 86, Birkhäuser Verlag Basel, 1988, pp. 477-493. Zbl0658.65142MR1022978
  14. Thomee V., Zhang N.-Y., Error estimates for semidiscrete finite element methods for parabolic integro-differential equations, Math. Comp. 53 (1989), 121-139. (1989) Zbl0673.65099MR0969493

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