Optimal error estimates for non-conforming approximations of linear parabolic problems with minimal regularity

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, nonconforming and mixed 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 nonconformity 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.