  We consider a general linear parabolic problem with extended time boundary
conditions (including initial value problems and periodic ones), and
approximate it by the implicit Euler scheme in time and the Gradient
Discretisation method in space; the latter is in fact a class of methods that
includes conforming and nonconforming finite elements, discontinuous Galerkin
methods and several others. The main result is an error estimate which holds
without supplementary regularity hypothesis on the solution. This result states
that the approximation error has the same order as the sum of the interpolation
error and the conformity error. The proof of this result relies on an inf-sup
inequality in Hilbert spaces which can be used both in the continuous and the
discrete frameworks. The error estimate result is illustrated by numerical
examples with low regularity of the solution.
