N. Yu. Bakaev, S. Larsson, and V. Thomee,

Long time behavior of backward difference type methods for
parabolic equations with memory in Banach space,

East-West J. Numer. Math. 6 (1998), 185-206.

Abstract.  We show stability in a Banach space framework of backward
Euler and second order backward difference timestepping methods for a
parabolic equation with memory.  The results are applied to derive
maximum norm stability estimates for piecewise linear finite element
approximations in a plane spatial domain, which is accomplished by a
new resolvent estimate for the discrete Laplacian.  Error estimates
are also given.

Keywords.  Integro-differential equation, parabolic, Banach space,
backward Euler method, backward difference method, sparse quadrature,
finite element method, maximum norm, resolvent estimate, exponential
decay.