On the discretization in time for a parabolic integrodifferential equation with a weakly singular kernel I: smooth initial data

摘要

Backward Euler methods are considered for a parabolic integrodifferential equation with a memory term containing a weakly singular kernel. The integral term is treated by means of a convolution quadrature suggested by Lubich. Some results from Lubich's discretized fractional calculus play a crucial role in error analyses. The stability and convergence properties of the time discretizations are derived and are applied to the semidiscrete equations built by the Galerkin finite element methods in the space variables.