N. Yu. Bakaev, S. Larsson, and V. Thom\'ee

Backward Euler type methods for parabolic integro-differential
equations in Banach space

RAIRO Mod\'el.~Math.~Anal.~Num\'er. 32 (1998), 85--99

Abstract: 
Time discretization by backward Euler type methods for a parabolic
equation with memory is studied.  Stability and error estimates are
proved under conditions that permit quadrature rules for approximation
of the memory term that have reduced storage requirements. The
analysis takes place in a Banach space framework, and the results are
used to derive error estimates in the $L_2$ and maximum norms for
piecewise linear finite element discretization in two space
dimensions.

Erratum: The printed version of the paper contains a minor error in
the beginning of the proof of Theorem 2.4.  This is corrected in the
version provided in this web-site. 

Address: 
N.~Yu.~Bakaev, 
Department of Mathematics, 
Air Force Engineering Academy, 
Leningradskii Prospekt 40, 
Moscow 125190,
Russia

E-mail: 
bakaev@postman.ru

Address:
S.~Larsson and V.~Thom\'ee, 
Department of Mathematics,          
Chalmers University of Technology and  G\"oteborg University,  
SE--412 96 G\"oteborg,               
Sweden  

E-mail: 
stig@math.chalmers.se, thomee@math.chalmers.se