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We consider a general linear parabolic problem with extended time boundaryconditions (including initial value problems and periodic ones), andapproximate it by the implicit Euler scheme in time and the GradientDiscretisation method in space; the latter is in fact a class of methods thatincludes conforming and nonconforming finite elements, discontinuous Galerkinmethods and several others. The main result is an error estimate which holdswithout supplementary regularity hypothesis on the solution. This result statesthat the approximation error has the same order as the sum of the interpolationerror and the conformity error. The proof of this result relies on an inf-supinequality in Hilbert spaces which can be used both in the continuous and thediscrete frameworks. The error estimate result is illustrated by numericalexamples with low regularity of the solution.

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