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We consider a singularly perturbed time-dependent problem with a shift termin space. On appropriately defined layer adapted meshes of Dur\'{a}n- andS-type we derive a-priori error estimates for the stationary problem. Using adiscontinuous Galerkin method in time we obtain error estimates for the fulldiscretisation. Introduction of a weighted scalar products and norms allows usto estimate the error of the time-dependent problem in energy and balancednorm. So far it was open to prove such a result. Error estimates in the energynorm is for the standard finite element discretization in space, and for theerror estimate in the balanced norm the computation of the numerical solutionis changed by using a different bilinear form. Some numerical results are givento confirm the predicted theory and to show the effect of shifts on thesolution.

A balanced norm error estimation for the time-dependent reaction-diffusion problem with shift in space