Long time behaviour of backward difference type methods for parabolic equations with memory in Banach space. (English) Zbl 0913.65139

Bakaev, N. Yu.; Larsson, S.; Thomée, V.

East-West J. Numer. Math. 6, No. 3, 185-206 (1998).

The behaviour of the solutions of the initial value problem \[ u_t+ Au= \int^t_0 b(t,\tau) Au(\tau)d\tau+ f(t),\quad t>0,\tag{\(*\)} \] with \(u(0)= u_0\), is studied in a Banach space framework. It is shown, under certain conditions, that \[ \| u(t)\|\leq CM(e^{-\sqrt t}\| u_0\|+ \int^t_0 e^{-\sqrt{(t-2)}}\| f(s)\| ds, \] \(t\geq 0\), where \(u(t)\) is a solution of \((*)\). The discrete analogue of \((*)\) is also studied. The results are applied to derive maximum norm stability estimates for piecewise linear finite element approximations in a plane spatial domain. Error estimates are given.

Reviewer: N.Parhi (Berhampur)

MSC:

65R20	Numerical methods for integral equations
45N05	Abstract integral equations, integral equations in abstract spaces
65J10	Numerical solutions to equations with linear operators (do not use 65Fxx)
45K05	Integro-partial differential equations

Keywords:

backward difference type methods; parabolic equations; integro-differential equation; finite element method; error estimates; Banach space; maximum norm stability