Nonsmooth Data Error Estimates

for Approximations of an Evolution Equation

with a Positive-Type Memory Term

Authors:
Ch. Lubich, I. H. Sloan and V. Thomée

Journal:
Math. Comp. **65** (1996), 1-17

MSC (1991):
Primary 45K05, 65M60, 65D32

DOI:
https://doi.org/10.1090/S0025-5718-96-00677-1

MathSciNet review:
1322891

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Abstract | References | Similar Articles | Additional Information

Abstract: We study the numerical approximation of an integro-differential equation which is intermediate between the heat and wave equations. The proposed discretization uses convolution quadrature based on the first- and second-order backward difference methods in time, and piecewise linear finite elements in space. Optimal-order error bounds in terms of the initial data and the inhomogeneity are shown for positive times, without assumptions of spatial regularity of the data.

**1**J.-C. López-Marcos,*A difference scheme for a nonlinear partial integrodifferential equation*, SIAM J. Numer. Anal.**27**(1990), 20--31. MR**91e:65160****2**C. Lubich,*Convolution quadrature and discretized operational calculus.*I, Numer. Math.**52**(1988), 129--145. MR**89g:65018****3**W. McLean and V. Thomée,*Numerical solution of an evolution equation with a positive type memory term*, J. Austral. Math. Soc. Ser. B**35**(1993), 23--70. MR**94e:65094****4**W. McLean, V. Thomée, and L.B. Wahlbin,*Discretization with variable time steps of an evolution equation with a positive type memory term*, Applied Mathematics Report, vol. AMR93/18, School of Mathematics, University of New South Wales.**5**O. Nevanlinna,*On the numerical solutions of some Volterra equations on infinite intervals*, Rev. Anal. Numér. Théorie Approximation**5**(1976), 31--57. MR**58:25043****6**J.M. Sanz-Serna,*A numerical method for a partial integro-differential equation*, SIAM J. Numer. Anal.**25**(1988), 319--327. MR**89d:65113****7**V. Thomée,*Galerkin finite element methods for parabolic problems*, Lecture Notes in Math., vol. 1054, Springer-Verlag, Berlin and New York, 1984. MR**86k:65006**

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Additional Information

**Ch. Lubich**

Affiliation:
Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany

Email:
lubich@na.mathematik.uni-tuebingen.de

**I. H. Sloan**

Affiliation:
School of Mathematics, University of New South Wales, Sydney 2052, Australia

Email:
I.Sloan@unsw.edu.au

**V. Thomée**

Affiliation:
Department of Mathematics, Chalmers University of Technology, S-412 96 Göteborg, Sweden

Email:
thomee@math.chalmers.se

DOI:
https://doi.org/10.1090/S0025-5718-96-00677-1

Keywords:
Evolution equation,
memory term,
nonsmooth data,
convolution quad\-ra\-ture

Received by editor(s):
August 30, 1994

Article copyright:
© Copyright 1996
American Mathematical Society